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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.).

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  • $\begingroup$ 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. $\endgroup$
    – user64494
    Feb 23, 2022 at 8:07
  • $\begingroup$ 2.02 seconds is a long time? $\endgroup$
    – 1729taxi
    Feb 23, 2022 at 10:39
  • $\begingroup$ 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. $\endgroup$
    – user64494
    Feb 23, 2022 at 10:41
  • $\begingroup$ Now that is slow - 26 seconds. $\endgroup$
    – 1729taxi
    Feb 23, 2022 at 10:48
  • $\begingroup$ It takes several minutes for me. $\endgroup$
    – user64494
    Feb 23, 2022 at 12:15

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