Track / Stage 8 / 123 of 180 #1823 of 2000

Problem 1823

IMO Shortlist mid-range; USAMO P2/P5

Geometry Difficulty 8.5 Prove it 62nd NMO SELECTION TESTS FOR THE BALKAN AND INTERNATIONAL MATHEMATICAL OLYMPIADS · Romania

Let $A_0A_1A_2$ be a non-equilateral triangle. The incircle of the triangle $A_0A_1A_2$ touches the side $A_iA_{i+1}$ at the point $T_{i+2}$ (indices are reduced modulo 3). Let $X_i$ be the perpendicular foot dropped from the point $T_i$ onto the line $T_{i+1}T_{i+2}$ . Show that the lines $A_iX_i$ are concurrent at a point situated on the Euler line of the triangle $T_0T_1T_2$ .

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

The lines $A_iA_{i+1}$ and $X_iX_{i+1}$ are parallel, for they are both antiparallel to the line $T_iT_{i+1}$ . Hence the triangles $A_0A_1A_2$ and $X_0X_1X_2$ are homologous: the three lines $A_iX_i$ are concurrent at the homology centre which lies on the homology line. The latter passes through the incentres of the two triangles: one is the circumcentre of the triangle $T_0T_1T_2$ and the other the orthocentre. The conclusion follows.

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