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.

$\mathrm{with}\left(\mathrm{geometry}\right)\:$

$\mathrm{point}\left(A\,0\,0\right),\mathrm{point}\left(B\,0\,3\right),\mathrm{point}\left(C\,1\,0\right),\mathrm{point}\left(H\,0\,6\right),\mathrm{point}\left(F\,2\,0\right)\:$

$\mathrm{point}\left(G\,3\,1\right)\:$

$\mathrm{triangle}\left(\mathrm{T1}\,\left[A\,B\,C\right]\right)\:$

$\mathrm{triangle}\left(\mathrm{T2}\,\left[A\,H\,F\right]\right)\:$

$\mathrm{triangle}\left(\mathrm{T3}\,\left[A\,H\,G\right]\right)\:$

$\mathrm{AreSimilar}\left(\mathrm{T1}\,\mathrm{T2}\right)$

$\mathrm{true}$

$\mathrm{AreSimilar}\left(\mathrm{T1}\,\mathrm{T3}\right)$

$\mathrm{false}$

$\mathrm{point}\left(H\,0\,\mathrm{Hy}\right),\mathrm{point}\left(G\,\mathrm{Gx}\,1\right)\:$

$\mathrm{AreSimilar}\left(\mathrm{T1}\,\mathrm{T3}\,'\mathrm{cond}'\right)$

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}}"

$\mathrm{FAIL}$

$\mathrm{cond}$

$\mathrm{\&or}\left(\left\{\frac{9}{{\mathrm{Hy}}^2}-\frac{10}{{\mathrm{Gx}}^2+{\left(\mathrm{Hy}-1\right)}^2}=0\,\frac{9}{{\mathrm{Hy}}^2}-\frac{1}{{\mathrm{Gx}}^2+1}=0\right\}\,\left\{\frac{9}{{\mathrm{Hy}}^2}-\frac{1}{{\mathrm{Gx}}^2+{\left(\mathrm{Hy}-1\right)}^2}=0\,\frac{9}{{\mathrm{Hy}}^2}-\frac{10}{{\mathrm{Gx}}^2+1}=0\right\}\,\left\{\frac{9}{{\mathrm{Gx}}^2+{\left(\mathrm{Hy}-1\right)}^2}-\frac{10}{{\mathrm{Hy}}^2}=0\,\frac{9}{{\mathrm{Gx}}^2+{\left(\mathrm{Hy}-1\right)}^2}-\frac{1}{{\mathrm{Gx}}^2+1}=0\right\}\,\left\{\frac{9}{{\mathrm{Gx}}^2+{\left(\mathrm{Hy}-1\right)}^2}-\frac{1}{{\mathrm{Hy}}^2}=0\,\frac{9}{{\mathrm{Gx}}^2+{\left(\mathrm{Hy}-1\right)}^2}-\frac{10}{{\mathrm{Gx}}^2+1}=0\right\}\,\left\{\frac{9}{{\mathrm{Gx}}^2+1}-\frac{10}{{\mathrm{Hy}}^2}=0\,\frac{9}{{\mathrm{Gx}}^2+1}-\frac{1}{{\mathrm{Gx}}^2+{\left(\mathrm{Hy}-1\right)}^2}=0\right\}\,\left\{\frac{9}{{\mathrm{Gx}}^2+1}-\frac{1}{{\mathrm{Hy}}^2}=0\,\frac{9}{{\mathrm{Gx}}^2+1}-\frac{10}{{\mathrm{Gx}}^2+{\left(\mathrm{Hy}-1\right)}^2}=0\right\}\right)$