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 Tips and Techniques: Maple Meets Marden's TheoremLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Robert J. LopezEmeritus Professor of Mathematics and Maple FellowMaplesoftLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The statement of Marden's theorem in Table 1 is taken from [1] where Dan Kalman gives insightful comments and an efficient geometric proof.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== be a third-degree polynomial with complex coefficients, and whose roots 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, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ are noncollinear points in the complex plane. Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be the triangle with vertices at 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, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+. There is a unique ellipse inscribed in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and tangent to the sides at their midpoints. The foci of this ellipse are the roots of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ6RidGL0YyRjlGOUY5.Table 1 Dan Kalman's statement of Marden's theorem
It is quite natural in Maple to start with three complex points, interpolate them with a (monic) cubic polynomial, and to find the zeros of its derivative. Then, from these two foci and the midpoints of the sides of the triangle whose vertices are the original three points, it is possible to obtain the equation of the inscribed ellipse. However, if the initial three points are taken as arbitrary complex points, the expressions that arise become exceedingly large and cumbersome.Kalman suggests the clever simplification of selecting the initial three points as 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, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ2RidGNEY3Rj5GPkY+, with the third point in the upper half-plane. There is no loss of generality by this device, since Kalman shows that under a linear transformation capturing translation, rotation, and scaling, Marden's theorem holds for a triple 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 if and only if it holds for the transformed triple. After observing how cumbersome the general expressions can become, we readily adopt the consequences of Kalman's insight.In Example 1 we start with three fixed points and use Maple's geometry package to obtain the inscribed ellipse. This is followed by an interactive animation that shows the dynamic dependence of the triangle, its midpoints, the inscribed ellipse, and its foci.In Example 2, we start with three points suggested by Kalman, the interpolating polynomial, and the zeros of its derivative as the foci. The indeterminates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= present grave problems for the geometry package, so we obtain the equation of the inscribed ellipse by interpolating a general quadratic through the midpoints of the sides of the triangle and having its center fall on the midpoint of the line segment connecting the foci. Tangency at the midpoints is then readily shown.Finally, for a constructive approach to Marden's theorem, we start with the triangle, and construct the ellipse tangent to its sides at the midpoints. The coordinates of the foci are obtained after transforming the ellipse to standard form. The foci so determined are shown to be the same as the zeros of the derivative of the polynomial interpolating the vertices of the triangle.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Example 1: A Specific Case</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Let the triple of complex numbers 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 be given by
The polynomial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== that interpolates these three points is
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The zeros of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ6RidGL0YyRjlGOUY5, namely, the foci of the inscribed ellipse, are then
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=It now becomes convenient to use Maple's geometry package for constructing the circumscribed ellipse. However, this requires that the vertices of the triangle, and the foci, be expressed as points in the package. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Define the three interpolated vertices as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY1LUYsNiVRIkNGJ0YvRjJGOQ==.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(); # input placeholderDefine the foci LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ as the geometry points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+.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, respectively the midpoints of sides LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI0JDRidGL0YyRjUtRiw2JVEjQUNGJ0YvRjJGOQ== of the triangle formed by the three interpolated vertices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY1LUYsNiVRIkNGJ0YvRjJGOQ==.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The coordinates of the three midpoints are found, and assigned the respective names 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.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(); # input placeholderDefine LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the triangle formed by the vertices 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}, as an object in the geometry package.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%;The ellipse command in the geometry package will define an ellipse from the foci and the distance from one focus to a point on the ellipse to the other focus. This distance is actually the length of the major axis. We take as the point on the ellipse the midpoint of one side of triangle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.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(); # input placeholderDefine (as an object in the geometry package) the ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to be the ellipse whose foci are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+, whose major axis has length LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and which contains the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI2FiRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw==.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEoZWxsaXBzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUYsNiVRIkVGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY2NiYtRiM2Ky1GLDYlUSVmb2NpRidGL0YyLUY+Ni1RIj1GJ0ZBRkMvRkdGRUZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGXm8tRjY2Ji1GIzYnLUklbXN1YkdGJDYmLUYsNiVRIkZGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjFGJ0ZBRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRidGPS1GZW82JkZnby1GIzYlLUZdcDYkUSIyRidGQUYvRjJGYHBGY3AvJStleGVjdXRhYmxlR0ZFRkFGQS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y9LUYsNiVRKWRpc3RhbmNlRidGL0YyRmluLUYsNiVRImRGJ0YvRjJGXXFGQUZBRl9xRmJxRj0tRjY2Ji1GIzYnLUYsNiVRInhGJ0YvRjJGPS1GLDYlUSJ5RidGL0YyRl1xRkFGQUZfcUZicUZdcUZBRkEtRj42LVEiOkYnRkFGQ0Zcb0ZIRkpGTEZORlBGXW9GX29GXXFGQQ==Obtain the Cartesian equation of ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, and assign this equation the name LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZXFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjZXFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSlFcXVhdGlvbkYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIkVGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUZaRjk=print(); # input placeholderLet LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be the center of ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Its coordinates are displayed to the right. Note the division by 3 for each coordinate. Careful inspection of the coordinates of the points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY1LUYsNiVRIkNGJ0YvRjJGOQ==, lead to the suspicion that center of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= falls on the centroid of triangle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.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print(); # input placeholderThis suspicion is confirmed by finding the coordinates of the centroid of triangle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEpY2VudHJvaWRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSlDZW50cm9pZEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiVEYnRi9GMi8lK2V4ZWN1dGFibGVHRkVGQUZBLUY+Ni1RIjpGJ0ZBRkMvRkdGRUZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGXG8tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGVC8lJmRlcHRoR0Zjby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GPjYtUSJ+RidGQUZDRmpuRkhGSkZMRk5GUEZSL0ZWRlQtRiw2JVEsY29vcmRpbmF0ZXNGJ0YvRjItRjY2JC1GIzYlRjpGZW5GQUZBRmVuRkE=print(); # input placeholderTable 2 The ellipse of Marden's theorem obtained with Maple's geometry package
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Figure 1, made with the draw command from the geometry package, shows the triangle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and the ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The foci LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ are shown as green dots, the center LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as a red dot, and the midpoints as blue dots. The vertices of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are drawn as black squares.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
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%;Figure 1 Circumscribed ellipse of Marden's theorem
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=That ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= touches the triangle at the midpoints of the sides is shown by the result of substituting the coordinates of the midpoints into the Cartesian equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= contains each midpoint since its equation is satisfied by each such point.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=To show that ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is tangent to the sides of the triangle at the midpoints, we need the derivative along the ellipse. This is obtained by Maple's implicitdiff command.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Table 3 compares the slope of each side of triangle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and the slope of ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at the midpoint of the corresponding side.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmc2xvcGVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJBRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJCRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ== = print(); # input 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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmc2xvcGVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJCRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJDRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ== = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmc2xvcGVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJBRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJDRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ== = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Table 3 Verification that ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is tangent to triangle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at the midpoints of its sides
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Apparently, taking as foci of an ellipse the zeros of the cubic polynomial interpolating the three vertices of a triangle in the complex plane, and forcing the ellipse to pass through the midpoint of one side of the triangle, is enough to determine an ellipse that is tangent to all three sides of the triangle at the midpoints of the sides.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Interactive Graphics</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Throughout the investigations detailed in this article, we tried to ignore the task of coding interactive Figure 2. The lure was too strong, however, so we first tried a coding based on the geometry package. Unfortunately, the resulting diagram responded too slowly to be useful. We then coded a solution based on interpolation: we interpolated a general quadratic through the midpoints of the triangle formed by three given points, using tangency to obtain the additional conditions needed to determine the six coefficients in the general quadratic. But the general quadratic is determined by just five coefficients - it is always possible to divide through by one of the nonzero coefficients.This distinction gives Maple's solve and fsolve commands some trouble. Since points are read from a graph as floating-point numbers, Maple's solve command finds just the zero solution of six equations in six unknowns that are not independent. Maple's fsolve command does not find any solution. However, if the floats are converted to rational numbers, solve will obtain the solution of the six equations in terms of one of the unknowns. By normalizing, we can then obtain the equation of the ellipse, and its graph can be added to a plot of the relevant points and lines.The code for Figure 2 is written as a module containing procedures that are invoked by the "click and drag" fields in the plot component. Initialize this code (and the figure) by clicking the button above Figure 2. It can be viewed by selecting Component Properties from the Context Menu for the button. The code is similar to what was used for the B\303\251zier Curves task template in the Curve Fitting section of the task templates.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=use DocumentTools, plots, Student[Precalculus] in
NEDRAM := module()
local L,N,dragIndex,q,a,b,c,d,e,f,qp;
export CL, DrawFig, DR, DragEnds;
q := a*x^2+b*x*y+c*y^2+d*x+e*y+f:
qp := implicitdiff(q,y,x):
Do(%Plot0 = plot([[0,0]], x=-5..5,y=0..10, style=point, axes=none, view=[-5..5,0..10], color=white)):
L := []:
N := 2;
dragIndex := -2;
############
CL := proc()
local cx,cy,G;
Do(cx = %Plot0(clickx));
Do(cy = %Plot0(clicky));
if nops(L)<N then
L := [L[],[cx,cy]];
G := plot(L,style=point, symbol=solidcircle, color=blue, symbolsize=15, view=[-5..5,0..10], axes=none);
Do(%Plot0 = G);
elif nops(L)=N then
L := [L[],[cx,cy]];
DrawFig(L);
dragIndex := -2;
else return;
end if;
end proc:
################
DrawFig := proc(L)
local LL,P,k,z,PP,F1,F2,H,M12,M23,M13,S12,S23,S13,eq,p,R,S,q1,q2,temp;
LL := convert(L,rational):
P := mul(z-(LL[k][1]+sqrt(-1)*LL[k][2]),k=1..3):
PP := evalc([solve(diff(P,z),z)]):
F1 := [Re(PP[1]),Im(PP[1])]:
F2 := [Re(PP[2]),Im(PP[2])]:
H := Midpoint(F1,F2):
M12 := Midpoint(LL[1],LL[2]):
M23 := Midpoint(LL[2],LL[3]):
M13 := Midpoint(LL[1],LL[3]):
S12 := Slope(LL[1],LL[2]):
S23 := Slope(LL[2],LL[3]):
S13 := Slope(LL[1],LL[3]):
eq[1] := eval(q,Equate([x,y],M12)):
eq[2] := eval(q,Equate([x,y],M23)):
eq[3] := eval(q,Equate([x,y],M13)):
eq[4] := numer(normal(eval(qp,Equate([x,y],M12))-S12)):
eq[5] := numer(normal(eval(qp,Equate([x,y],M23))-S23)):
eq[6] := numer(normal(eval(qp,Equate([x,y],M13))-S13)):
S := solve({seq(eq[k],k=1..6)},{a,b,c,d,e,f}):
q1 := eval(q,S):
temp := coeff(q1,x,2):
q2 := simplify(q1/temp):
p[1] := implicitplot(q2,x=-5..5,y=0..10,gridrefine=3,rangeasview):
p[2] := plot([[LL[1],LL[2]],[LL[2],LL[3]],[LL[1],LL[3]]],color=black):
p[3] := plot([M12,M23,M13],style=point,symbol=solidcircle,color=blue,symbolsize=15):
p[4] := plot(LL,style=point,symbol=solidbox,color=black,symbolsize=15):
p[5] := plot([H],style=point,symbol=solidcircle,color=red,symbolsize=15):
p[6] := plot([F1,F2],style=point,symbol=solidcircle,color=green,symbolsize=15):
R := display([seq(p[k],k=1..6)],axes=none,scaling=constrained):
Do(%Plot0 = R);
end proc:
##################
DR := proc()
local x0,y0, x1,y1,j;
Do(x0 = %Plot0(startx));
Do(y0 = %Plot0(starty));
Do(x1 = %Plot0(endx));
Do(y1 = %Plot0(endy));
if abs(x1)>5 then x1:=5*signum(x1);end if;
if y1>10 then y1:=10;end if;
if y1<0 then y1:=0;end if;
if dragIndex < 0 then
for j to nops(L) do
if Student[Precalculus][Distance](L[j],[x0,y0])<.8 then
dragIndex := j; end if;
end do;
else L := eval(L,L[dragIndex]=[x1,y1]); DrawFig(L);
end if;
end proc:
########################
DragEnds := proc()
dragIndex := -2;
end proc:
end module:
end use: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=use DocumentTools in
if type(NEDRAM,`module`) then NEDRAM:-DragEnds() end if;
end use;
use DocumentTools in
if type(NEDRAM,`module`) then NEDRAM:-CL() end if;
end use;
use DocumentTools in
if type(NEDRAM,`module`) then NEDRAM:-DR() end if;
end use;
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2 Interactive investigation of Marden's theorem. Click to place three points. Third click generates the full diagram of ellipse and triangle. Drag any vertex to resize the triangle and the inscribed ellipse.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=If the cursor on the graph does not default to the "click and drag" probe, select it from the toolbar.
<Text-field style="Heading 1" layout="Heading 1">Example 2: Using Kalman's Simplification</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=If all three vertices are taken as arbitrary complex points, there are six indeterminates in any investigation of Marden's theorem. Kalman showed that it suffices to take the vertices 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 as the complex points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYzNi1RKiZ1bWludXMwO0YnRi9GNi9GOkY4RjxGPkZARkJGRC9GR1EsMC4yMjIyMjIyZW1GJy9GSkZRRitGMkYv and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJpRidGL0YyLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVi1GLDYlUSJ2RidGL0YyRjk=, thereby reducing the number of indeterminates to a more tractable two. The interpolating polynomial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is then
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(); # input 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(); # input placeholderIts derivative,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print(); # input 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=has the two 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(); # input placeholderthe Cartesian coordinates of which areLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZgcC1JI21pR0YkNiVRJ3RlbXAxckYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJmV2YWxjRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEjUmVGJy9GMEY9RjktRlM2JC1GIzYmLUYsNiVRIlpGJ0YvRjItRlM2Ji1GIzYlLUkjbW5HRiQ2JFEiMUYnRjkvJStleGVjdXRhYmxlR0Y9RjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zkb0Y5RjlGZG9GOUY5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GYHAtRjY2LVEpYXNzdW1pbmdGJ0Y5RjtGPkZARkJGREZGRkhGX3BGYXBGXHAtRiw2JVEidkYnRi9GMi1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GYW82JFEiMEYnRjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGYHAvJSZkZXB0aEdGZnEvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRmJxNiZGZHFGZ3FGaXEvRlxyUSVhdXRvRictRiw2JVEndGVtcDJyRidGL0YyRjVGTy1GUzYkLUYjNiZGVy1GUzYkLUYjNiZGaW4tRlM2Ji1GIzYlLUZhbzYkUSIyRidGOUZkb0Y5RjlGZm9GaW9GZG9GOUY5RmRvRjlGOUZccEZicEZccEZlcEZocEZbcUZecUZhcUZeci1GLDYlUSd0ZW1wMWlGJ0YvRjJGNUZPLUZTNiQtRiM2Ji1GLDYlUSNJbUYnRlpGOUZlbkZkb0Y5RjlGXHBGYnBGXHBGZXBGaHBGW3EtRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZfcC9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ1RidGL0YyRmhwRltxRl5xRmFxRl5yLUYsNiVRJ3RlbXAyaUYnRi9GMkY1Rk8tRlM2JC1GIzYmRlt0RmlyRmRvRjlGOUZccEZicEZccEZlcEZocEZbcUZedEZkdEZocEZbcUZecUZhcUZccC1JJW1zdWJHRiQ2Ji1GLDYlUSJmRidGL0YyLUYjNidGYG9GL0Zkby8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0YyLyUvc3Vic2NyaXB0c2hpZnRHRl1xL0krbXNlbWFudGljc0dGJFEnYXRvbWljRidGNS1GUzYmLUYjNidGK0ZedEZkc0Zkb0Y5RjlGZm9GaW9GXnFGYXFGXHAtRl91NiZGYXUtRiM2J0Zhc0YvRmRvRmZ1RjJGaXVGW3ZGNS1GUzYmLUYjNidGYnJGXnRGZ3RGZG9GOUY5RmZvRmlvRl5xRmFxRlxwLUZfdTYlRmF1RmR1Rml1LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNRl51LUY2Ni1RIjtGJ0Y5RjtGYXRGQEZCRkRGRkZIRl9wRk1GYXFGXHAtRl91NiVGYXVGZHZGaXVGXHdGYnZGXHBGZG9GOQ==print(); # input placeholder
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Table 4 summarizes the ensuing calculations in the geometry package.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
As we did earlier, define the three interpolated vertices as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY1LUYsNiVRIkNGJ0YvRjJGOQ==.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print(); # input placeholderDefine the foci LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ as the geometry points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+.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, respectively the midpoints of sides LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjQUJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI0JDRidGL0YyRjUtRiw2JVEjQUNGJ0YvRjJGOQ== of the triangle formed by the three interpolated vertices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQkYnRi9GMkY1LUYsNiVRIkNGJ0YvRjJGOQ==.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The coordinates of the three midpoints are found, and assigned the respective names 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.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(); # input placeholderThe center of the circumscribed ellipse will be at the midpoint of the segment joining the foci. Its coordinates are again those of the centroid of the triangle whose vertices are 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.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print(); # input placeholderThe distance from one focus to a point on the ellipse to the other focus 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print(); # input placeholderUnfortunately, this expression for the length of the major axis is too complex for the ellipse command in the geometry package to handle. It is not possible to construct ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as we did in Table 2. We are forced to follow another path.Table 4 Use of Maple's geometry package to find the ellipse of Marden's theorem in terms of the parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The ellipse with foci 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 has the known center 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. The coordinates of the center of the conic governed by the quadratic equation
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Since at least one of the six coefficients in the general quadratic must be nonzero, the ellipse in Marden's theorem is determined by five constants. Interpolating the quadratic through the three midpoints of the sides of the circumscribing triangle provides three conditions, and prescribing the center provides two more. Hence, the solution of the five equations
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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=isLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJnNvbHZlRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNictRlM2Ji1GIzYmLUYsNiVRJHNlcUYnRi9GMi1GUzYkLUYjNistSSVtc3ViR0YkNiUtRiw2JVEicUYnRi9GMi1GIzYlLUYsNiVRImtGJ0YvRjJGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJ0Zkby1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORl9wLUZmcDYkUSI1RidGOS8lK2V4ZWN1dGFibGVHRj1GOUY5RmJxRjlGOS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGam8tRlM2Ji1GIzYtLUYsNiVRImFGJ0YvRjJGam8tRiw2JVEiYkYnRi9GMkZqby1GLDYlUSJjRidGL0YyRmpvLUYsNiVRImRGJ0YvRjJGam8tRiw2JVEiZkYnRi9GMkZicUY5RjlGZHFGZ3FGYnFGOUY5RmJxRjk=print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=so the ellipse is described by the 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(); # input placeholder
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=where we have divided through by the parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. To parallel the verifications in Table 3, we first obtain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGL0YyRjlGOUY5 implicitly as
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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = print(); # input 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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Table 5 Verification that at the midpoints of the sides of the enveloping triangle, the ellipse of Marden's theorem is tangent to the triangle
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Executing the command
generates (as a pop-up) the interactive plotting tool whose screen-capture is seen in Figure 3.
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3 Interactive plot of the ellipse in Marden's theorem
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This same tool can be launched from the Plot Builder by selecting "Interactive Plot with 2 parameters" as the plot type.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Marden's Theorem by Computation</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=By rotation and translation, but not scaling, the three vertices 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 of a triangle can be moved to the points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRIjBGJ0Y+Rj5GPkY6LUYsNiQtRiM2Jy1GOzYtUSomdW1pbnVzMDtGJ0Y+RkAvRkRGQkZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGXG9GMEY6RlVGPkY+RjotRiw2JC1GIzYmLUYxNiVRInVGJ0Y0RjdGOi1GMTYlUSJ2RidGNEY3Rj5GPkY+ in the plane. We next establish that the foci of an ellipse that is tangent to the sides of the triangle at its midpoints are the zeros of the derivative of the polynomial interpolating the vertices.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We start with the general quadratic equation
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The midpoints and slopes of the sides of the triangle formed by the vertices 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 are listed in Table 6.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Table 6 Midpoints (left column) and slopes (right column) of the sides of the triangle whose vertices are 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
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The six coefficients in are determined by the equations
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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=whose solution 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print(); # input placeholderleading to the quadraticLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Using , we obtain the coordinates of the center as
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=in keeping with our earlier observation about the center of the ellipse being at the centroid of the triangle. If we next translate coordinates 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=the quadratic becomes
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEjV1RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUShjb2xsZWN0RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiotRiw2JVElZXZhbEYnRi9GMi1GUzYkLUYjNictRiw2JVEiV0YnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUZTNiYtRiM2My1GLDYlUSJ4RidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRIlhGJ0YvRjItRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZkcC1GLDYlUSJ1RidGL0YyLUY2Ni1RIi9GJ0Y5RjtGPi9GQUYxRkJGREZGRkgvRktRLDAuMTY2NjY2N2VtRicvRk5GXnEtSSNtbkdGJDYkUSIzRidGOUZbby1GLDYlUSJ5RidGL0YyRmpvLUYsNiVRIllGJ0YvRjJGYHAtRiw2JVEidkYnRi9GMkZpcEZgcS8lK2V4ZWN1dGFibGVHRj1GOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRl1yRjlGOUZbby1GUzYmLUYjNidGXXBGW29GZ3FGXXJGOUY5Rl9yRmJyRltvLUYsNiVRKXNpbXBsaWZ5RidGL0YyRl1yRjlGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zgby8lJmRlcHRoR0Zkcy8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRl9vL0ZORmBvRitGam8tRmFxNiRRIjBGJ0Y5Rl1yRjk=print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=the template for which 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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=If the coordinates are rotated 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkdGFuRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMkYnRjItSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItRiw2JVEnJiM5NTI7RidGL0YyRjJGMi1GPzYtUSI9RidGMkZCRkRGRkZIRkpGTEZOL0ZRUSwwLjI3Nzc3NzhlbUYnL0ZURmZuLUkmbWZyYWNHRiQ2KC1GLDYlUSJCRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2Jy1GLDYlUSJBRidGXm9GYG8tRj82LVEoJm1pbnVzO0YnRjJGQkZERkZGSEZKRkxGTi9GUVEsMC4yMjIyMjIyZW1GJy9GVEZbcC1GLDYlUSJDRidGXm9GYG9GXm9GYG8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmVwLyUpYmV2ZWxsZWRHRjFGMg==, the quadratic assumes the form
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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Setting 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 equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, we getLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Unfortunately, Maple is unable to recognize that the coefficients of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjU= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5NTE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjU= can be further simplified so that 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orLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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orLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiQ0YnRi9GMkY5, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are positive, the coefficient of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5NTE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjU= is smaller than the coefficient of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjU=. Hence, the standard form for this ellipse is 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 and 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Therefore we haveLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Since 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, and since 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 for an ellipse, the denominator for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= is positive, so that 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=In order to show that the foci of the ellipse are the same as
(that is, the zeros of derivative of the interpolating cubic polynomial), we rotate and translate the points 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 back to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-coordinate system where the foci should then be 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. The most difficult part of the rotation is expressing sine and cosine of 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 in terms of radicals. For the sine we 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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=and make the substitutionsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEqc2luZWhhbGZURicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVElZXZhbEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRJXRlbXBGJ0YvRjItRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GUzYmLUYjNjYtRiw2JVEiVUYnRi9GMi1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JJW1zdXBHRiQ2JS1GLDYlUSJ2RidGL0YyLUYjNictSSNtbkdGJDYkUSIyRidGOUYvLyUrZXhlY3V0YWJsZUdGPS8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORl5xLUZnbzYlLUYsNiVRInVGJ0YvRjJGXHBGZ3BGanAtRl9wNiRRIjNGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGaG4vRk5GaW4tRmdvNiUtRiw2JVEickYnRi9GMkZccEZncEZaLUYsNiVRIlZGJ0YvRjJGY28tRjY2LVEqJnVtaW51czA7RidGOUY7Rj5GQEZCRkRGRkZIRl1xRl9xRl5wRmhxRmJxRmhxRmlvRmJwRjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZicEY5RjlGaHFGYnBGOQ==print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Expressing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEidkYnRi9GMkY5, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, we 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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The rotation and translation of the first coordinate of the first focus is thenLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVElbmV3Y0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRKXNpbXBsaWZ5RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiwtRjY2L1EqJnVtaW51czA7RicvJStleGVjdXRhYmxlR0Y9LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZqbi1GLDYnUSpzaW5laGFsZlRGJ0YvRlpGZm5GMi1GNjYvUScmc2RvdDtGJ0ZaRmZuRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZjby1GLDYnUScmIzk1NTtGJy9GMEY9RlpGZm5GOS1GNjYvUSIrRidGWkZmbkY5RjtGPkZARkJGREZGRkhGaW5GW28tRiw2J1EidUYnRi9GWkZmbkYyLUY2Ni9RIi9GJ0ZaRmZuRjlGO0Y+L0ZBRjFGQkZERkZGSC9GS1EsMC4xNjY2NjY3ZW1GJy9GTkZkcC1JI21uR0YkNiZRIjNGJ0ZaRmZuRjlGWkY5RjktRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZib0Zkby1GNjYtUSlhc3N1bWluZ0YnRjlGO0Y+RkBGQkZERkZGSEZib0Zkb0ZqcC1GLDYlUSJyRidGL0YyLUY2Ni1RIj5GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUZncDYkUSIwRidGOUZaRjk=print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The equivalence to the first coordinate of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ is shown by the vanishing of the difference:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We leave three similar calculations to the reader.
<Text-field style="Heading 1" layout="Heading 1">Note on Hidden Code</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Cells tinted in yellow contain hidden Maple input. This input can be seen by checking the "Show input" box near the bottom of the Table Properties dialog, accessed either through the Table menu, or the Context Menu. Figure 1 and Table 6 also contain hidden Maple input, but are not tinted.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">References</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
[1]Dan Kalman, "An Elementary Proof of Marden's Theorem," American Mathematical Monthly, Volume 115, Number 4, April 2008, pp 330-338.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Legal Notice: \302\251 Maplesoft, a division of Waterloo Maple Inc. 2010. Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=