Gergonne and Medial Triangles Are Orthologic: What is this about?
A Mathematical Droodle

Explanation

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In ΔABC, let M_a, M_b, M_c be the midpoints of sides BC, AC, and AB. Let A', B', C' be the points where the incircle touches those sides. ( ΔA'B'C' is known as Gergonne or contact triangle of ΔABC.) Then the perpendiculars from the midpoints M_a, M_b, M_c onto the opposite sides of ΔA'B'C' are concurrent.

According to R. Honsberger, the problem has been submitted by Bulgaria to an IMO, but was left unused. It was discussed and solved by J. T. Groenman, The Netherlands, in Crux Mathematicorum, 1987, 75. Here's Groenman's solution:

The center of the incircle of a triangle lies at the intersection of its angle bisectors. The incenter is thus naturally equidistant from the sides of the triangle. It follows that the tangents AB', AC' from vertex A of ΔABC to its incircle are equal and ΔAB'C' is isosceles. The bisector of angle A is perpendicular to B'C', hence parallel to the perpendicular from M_a. The latter, being parallel to the angle bisector in ΔABC, serves as the angle bisector of angle M_bM_aM_c in the medial triangle ΔM_aM_bM_c, since the two triangles are homothetic. The same holds for the other two perpendiculars. The three intersect as being angle bisectors of ΔM_aM_bM_c.

The fact just proven means that the Gergonne and medial triangles are orthologic, which, as we know from Maxwell's theorem, is a symmetric relationship. We conclude that the perpendiculars from the vertices of Gergonne triangle A'B'C' onto the opposite sides of ΔM_aM_bM_c also concur. But this fact is obvious. Indeed the sides of the medial triangle are parallel to those of ΔABC, so that the perpendiculars at the vertices of ΔA'B'C' where the incircle of ΔABC touches its sides meet at the incenter of ΔABC, naturally. Going backwards, we get another proof of the Bulgarian problem.

References

1. R. Honsberger, From Erdös To Kiev, MAA, 1996, pp. 99-102

1. Gergonne and Medial Triangles Are Orthologic
2. Pedal Triangle and Isogonal Conjugacy
3. Orthologic Triangles in a Quadrilateral

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