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Problem 813

AMC 12 late, AIME early

Geometry Difficulty 4.4 Prove it Belarus — Selection and Training Session · 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 $CH$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official 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.

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