Competition problem — Geometry — Maths Olympiad Prep

Library / /55 of 70

Geometry Difficulty 8.5 Shortlist Prove it 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 $T_0T_1T_2$ .

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.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.