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 GergonnePoint
 find the Gergonne point of a given triangle

 Calling Sequence GergonnePoint(G, ABC)

Parameters

 G - the name of the Gergonne point ABC - triangle

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{geometry}\right):$
 > $\mathrm{triangle}\left(T,\left[\mathrm{point}\left(A,0,0\right),\mathrm{point}\left(B,2,0\right),\mathrm{point}\left(C,1,3\right)\right]\right):$
 > $\mathrm{GergonnePoint}\left(G,T\right)$
 ${G}$ (1)
 > $\mathrm{detail}\left(G\right)$
 $\begin{array}{ll}{\text{name of the object}}& {G}\\ {\text{form of the object}}& {\mathrm{point2d}}\\ {\text{coordinates of the point}}& \left[{1}{,}\frac{{3}{}\left({10}{+}\sqrt{{10}}\right)}{{19}{}\sqrt{{10}}{+}{10}}\right]\end{array}$ (2)

draw the picture of the above definition for the triangle T

 > $\mathrm{incircle}\left(c,T\right):$
 > $\mathrm{segment}\left(\mathrm{sg1},A,\mathrm{projection}\left(H,\mathrm{center}\left(c\right),\mathrm{line}\left(\mathrm{tmp},\left[B,C\right]\right)\right)\right):$
 > $\mathrm{segment}\left(\mathrm{sg2},B,\mathrm{projection}\left(E,\mathrm{center}\left(c\right),\mathrm{line}\left(\mathrm{tmp},\left[C,A\right]\right)\right)\right):$
 > $\mathrm{segment}\left(\mathrm{sg3},C,\mathrm{projection}\left(F,\mathrm{center}\left(c\right),\mathrm{line}\left(\mathrm{tmp},\left[A,B\right]\right)\right)\right):$
 > $\mathrm{draw}\left(\left\{\mathrm{sg1},\mathrm{sg2},\mathrm{sg3},G\left(\mathrm{symbol}=\mathrm{DIAMOND}\right),T\left(\mathrm{color}=\mathrm{red}\right),c\left(\mathrm{color}=\mathrm{green},\mathrm{style}=\mathrm{POINT}\right)\right\},\mathrm{color}=\mathrm{blue},\mathrm{printtext}=\mathrm{true}\right)$