Let H, E, and F be the points of contact of the inscribed circle of triangle ABC with the sides BC, CA, AB respectively. AH, BE, CF are concurrent.

The point of concurrency is called the Gergonne point of the triangle, after J. D. Gergonne (1771-1859), founder-editor of the mathematics journal Annales de mathematiques. Just why the point was named after Gergonne is not known.

For a detailed description of the Gergonne point G, use the routine detail (i.e., detail(G))

Note that the routine only works if the vertices of the triangle are known.

The command with(geometry,GergonnePoint) allows the use of the abbreviated form of this command.

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

$\mathrm{triangle}\left(T\,\left[\mathrm{point}\left(A\,0\,0\right)\,\mathrm{point}\left(B\,2\,0\right)\,\mathrm{point}\left(C\,1\,3\right)\right]\right)\:$

$\mathrm{GergonnePoint}\left(G\,T\right)$

$G$

$\mathrm{detail}\left(G\right)$

$\begin{array}{ll}\text{name of the object}& G\\ \text{form of the object}& \mathrm{point2d}\\ \text{coordinates of the point}& \left[1\,\frac{3\left(10+\sqrt{10}\right)}{19\sqrt{10}+10}\right]\end{array}$

$\mathrm{incircle}\left(c\,T\right)\:$

$\mathrm{segment}\left(\mathrm{sg1}\,A\,\mathrm{projection}\left(H\,\mathrm{center}\left(c\right)\,\mathrm{line}\left(\mathrm{tmp}\,\left[B\,C\right]\right)\right)\right)\:$

$\mathrm{segment}\left(\mathrm{sg2}\,B\,\mathrm{projection}\left(E\,\mathrm{center}\left(c\right)\,\mathrm{line}\left(\mathrm{tmp}\,\left[C\,A\right]\right)\right)\right)\:$

$\mathrm{segment}\left(\mathrm{sg3}\,C\,\mathrm{projection}\left(F\,\mathrm{center}\left(c\right)\,\mathrm{line}\left(\mathrm{tmp}\,\left[A\,B\right]\right)\right)\right)\:$

$\mathrm{draw}\left(\left\{\mathrm{sg1}\,\mathrm{sg2}\,\mathrm{sg3}\,G\left(\mathrm{symbol}=\mathrm{DIAMOND}\right)\,T\left(\mathrm{color}=\mathrm{red}\right)\,c\left(\mathrm{color}=\mathrm{green}\,\mathrm{style}=\mathrm{POINT}\right)\right\}\,\mathrm{color}=\mathrm{blue}\,\mathrm{printtext}=\mathrm{true}\right)$