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centroid of triangle when line varies

Source: French MO 1995 P1

April 22, 2021
geometryTriangle

Problem Statement

We are given a triangle ABCABC in a plane PP. To any line DD, not parallel to any side of the triangle, we associate the barycenter GDG_D of the set of intersection points of DD with the sides of △ABC\triangle ABC. The object of this problem is determining the set F\mathfrak F of points GDG_D when DD varies.
(a) If DD goes over all lines parallel to a given line δ\delta, prove that GDG_D describes a line Δδ\Delta_\delta. (b) Assume △ABC\triangle ABC is equilateral. Prove that all lines Δδ\Delta_\delta are tangent to the same circle as δ\delta varies, and describe the set F\mathfrak F. (c) If ABCABC is an arbitrary triangle, prove that one can find a plane PP and an equilateral triangle A′B′C′A'B'C' whose orthogonal projection onto PP is △ABC\triangle ABC, and describe the set F\mathfrak F in the general case.
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