Let H, E, F be the points on the sides BC, CA, AB of triangle ABC such that H is half way around the perimeter from A, E half way around from B, and F half way around from C. AH, BE, CF are concurrent. This point of concurrence is called the Nagel point of the triangle, after C. H. Nagel (1803-1882).

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

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

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

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

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

$\mathrm{NagelPoint}\left(N\,T\right)$

$N$

$\mathrm{coordinates}\left(N\right)$

$\left[1\,\frac{6\sqrt{10}-3\sqrt{4}}{2\sqrt{10}+\sqrt{4}}\right]$