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 AreSimilar
 test if two triangles are similar

 Calling Sequence AreSimilar(T1, T2, cond)

Parameters

 T1, T2 - two triangles cond - (optional) name

Description

 • Two similar triangles T1 and T2 are triangles whose corresponding angles are congruent and whose corresponding sides are in proportion.
 • The routine returns true if T1 and T2 are similar; false if they are not; and FAIL if it is unable to reach a conclusion.
 • In FAIL is returned, and the optional argument is given, the condition that makes T1 and T2 similar is assigned to this argument. It will be either of the form $\mathrm{expr}=0$ or of the form $&\mathrm{or}\left(\mathrm{expr_1}=0,...,\mathrm{expr_n}=0\right)$ where expr, expr_i are Maple expressions.
 • The command with(geometry,AreSimilar) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{geometry}\right):$
 > $\mathrm{point}\left(A,0,0\right),\mathrm{point}\left(B,0,3\right),\mathrm{point}\left(C,1,0\right),\mathrm{point}\left(H,0,6\right),\mathrm{point}\left(F,2,0\right):$
 > $\mathrm{point}\left(G,3,1\right):$
 > $\mathrm{triangle}\left(\mathrm{T1},\left[A,B,C\right]\right):$
 > $\mathrm{triangle}\left(\mathrm{T2},\left[A,H,F\right]\right):$
 > $\mathrm{triangle}\left(\mathrm{T3},\left[A,H,G\right]\right):$
 > $\mathrm{AreSimilar}\left(\mathrm{T1},\mathrm{T2}\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{AreSimilar}\left(\mathrm{T1},\mathrm{T3}\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{point}\left(H,0,\mathrm{Hy}\right),\mathrm{point}\left(G,\mathrm{Gx},1\right):$
 > $\mathrm{AreSimilar}\left(\mathrm{T1},\mathrm{T3},'\mathrm{cond}'\right)$
 AreSimilar:   "hint: one of the following conditions must be satisfied: {{9/Hy^2-10/(Gx^2+(Hy-1)^2) = 0, 9/Hy^2-1/(Gx^2+1) = 0}, {9/Hy^2-1/(Gx^2+(Hy-1)^2) = 0, 9/Hy^2-10/(Gx^2+1) = 0}, {9/(Gx^2+(Hy-1)^2)-10/Hy^2 = 0, 9/(Gx^2+(Hy-1)^2)-1/(Gx^2+1) = 0}, {9/(Gx^2+(Hy-1)^2)-1/Hy^2 = 0, 9/(Gx^2+(Hy-1)^2)-10/(Gx^2+1) = 0}, {9/(Gx^2+1)-10/Hy^2 = 0, 9/(Gx^2+1)-1/(Gx^2+(Hy-1)^2) = 0}, {9/(Gx^2+1)-1/Hy^2 = 0, 9/(Gx^2+1)-10/(Gx^2+(Hy-1)^2) = 0}}"
 ${\mathrm{FAIL}}$ (3)
 > $\mathrm{cond}$
 $\left(\left\{\frac{{9}}{{{\mathrm{Hy}}}^{{2}}}{-}\frac{{10}}{{{\mathrm{Gx}}}^{{2}}{+}{\left({\mathrm{Hy}}{-}{1}\right)}^{{2}}}{=}{0}{,}\frac{{9}}{{{\mathrm{Hy}}}^{{2}}}{-}\frac{{1}}{{{\mathrm{Gx}}}^{{2}}{+}{1}}{=}{0}\right\}\right){\vee }\left(\left\{\frac{{9}}{{{\mathrm{Hy}}}^{{2}}}{-}\frac{{1}}{{{\mathrm{Gx}}}^{{2}}{+}{\left({\mathrm{Hy}}{-}{1}\right)}^{{2}}}{=}{0}{,}\frac{{9}}{{{\mathrm{Hy}}}^{{2}}}{-}\frac{{10}}{{{\mathrm{Gx}}}^{{2}}{+}{1}}{=}{0}\right\}\right){\vee }\left(\left\{\frac{{9}}{{{\mathrm{Gx}}}^{{2}}{+}{\left({\mathrm{Hy}}{-}{1}\right)}^{{2}}}{-}\frac{{10}}{{{\mathrm{Hy}}}^{{2}}}{=}{0}{,}\frac{{9}}{{{\mathrm{Gx}}}^{{2}}{+}{\left({\mathrm{Hy}}{-}{1}\right)}^{{2}}}{-}\frac{{1}}{{{\mathrm{Gx}}}^{{2}}{+}{1}}{=}{0}\right\}\right){\vee }\left(\left\{\frac{{9}}{{{\mathrm{Gx}}}^{{2}}{+}{\left({\mathrm{Hy}}{-}{1}\right)}^{{2}}}{-}\frac{{1}}{{{\mathrm{Hy}}}^{{2}}}{=}{0}{,}\frac{{9}}{{{\mathrm{Gx}}}^{{2}}{+}{\left({\mathrm{Hy}}{-}{1}\right)}^{{2}}}{-}\frac{{10}}{{{\mathrm{Gx}}}^{{2}}{+}{1}}{=}{0}\right\}\right){\vee }\left(\left\{\frac{{9}}{{{\mathrm{Gx}}}^{{2}}{+}{1}}{-}\frac{{10}}{{{\mathrm{Hy}}}^{{2}}}{=}{0}{,}\frac{{9}}{{{\mathrm{Gx}}}^{{2}}{+}{1}}{-}\frac{{1}}{{{\mathrm{Gx}}}^{{2}}{+}{\left({\mathrm{Hy}}{-}{1}\right)}^{{2}}}{=}{0}\right\}\right){\vee }\left(\left\{\frac{{9}}{{{\mathrm{Gx}}}^{{2}}{+}{1}}{-}\frac{{1}}{{{\mathrm{Hy}}}^{{2}}}{=}{0}{,}\frac{{9}}{{{\mathrm{Gx}}}^{{2}}{+}{1}}{-}\frac{{10}}{{{\mathrm{Gx}}}^{{2}}{+}{\left({\mathrm{Hy}}{-}{1}\right)}^{{2}}}{=}{0}\right\}\right)$ (4)

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