# The Nagel point revisited

Let $$i$$ be the incenter of the triangle $$\triangle abc$$, and $$n$$ be the Nagel point of this triangle (see Wiki ). Let also $$h_3$$ and $$h_2$$ be the orthocenters of the triangles $$\triangle iab$$ and $$\triangle iac$$, respectively. Is it possible to prove with GeometricScene that the lines $$an$$ and $$h_2h_3$$ intersect on the inscribed circle of the triangle $$\triangle abc$$?

My first step is

RandomInstance[GeometricScene[{a, b, c, i}, {Triangle[{a, b, c}],
Insphere[{a, b, c}], i == TriangleCenter[{a, b, c}, "Incenter"],
Triangle[{i, a, b}], Triangle[{i, a, c}]}]]


which works well (Don't know how to insert a picture produced by the above code here.).

• RandomInstance[ GeometricScene[{a, b, c, i, h2, h3}, {Triangle[{a, b, c}], Insphere[{a, b, c}], i == TriangleCenter[{a, b, c}, "Incenter"], Triangle[{i, a, b}], Triangle[{i, a, c}], h2 == TriangleCenter[{a, b, i}, "Orthocenter"], h3 == TriangleCenter[{a, c, i}, "Orthocenter"]}]] works too. It takes a long time. Feb 23, 2022 at 8:07
• 2.02 seconds is a long time? Feb 23, 2022 at 10:39
• My comp is not strong. RandomInstance[ GeometricScene[{a, b, c, i, h2, h3}, {Triangle[{a, b, c}], Insphere[{a, b, c}], i == TriangleCenter[{a, b, c}, "Incenter"], Triangle[{i, a, b}], Triangle[{i, a, c}], h2 == TriangleCenter[{a, b, i}, "Orthocenter"], h3 == TriangleCenter[{a, c, i}, "Orthocenter"], InfiniteLine[{h2, h3}]}]] also works. Feb 23, 2022 at 10:41
• Now that is slow - 26 seconds. Feb 23, 2022 at 10:48
• It takes several minutes for me. Feb 23, 2022 at 12:15