Competition problem — Geometry — Maths Olympiad Prep

Library / /1 of 35

Geometry Difficulty 4.4 AIME Prove it Belarus

The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$ .
If $\omega_1$ touches the altitude $CH$ prove that $\omega_2$ also touches $CH$ .

Solution

The statement easily follows from the following fact: if points $A_1$ , $B_1$ are chosen on the sides $AC$ , $BC$ of the triangle $ABC$ , respectively, then the radical axis of the circles with the diameters $AA_1$ and $BB_1$ contains the orthocenter of the triangle $ABC$ .

In case when $AA_1$ and $BB_1$ are medians the centerline of $\omega_1$ and $\omega_2$ is obviously parallel to the side $AB$ , then the radical axis is perpendicular to $AB$ , thus the statement 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 and solution reproduced as published; topic and difficulty added by this site.