Problem

Source: Bolivia IMO TST 2022 Day 2 P1

Tags: geometry, circumcircle, Bolivia, TST, midpoints



On $\triangle ABC$, let $M$ the midpoint of $AB$ and $N$ the midpoint of $CM$. Let $X$ a point such that $\angle XMC=\angle MBC$ and $\angle XCM=\angle MCB$ with $X,B$ in opposite sides of line $CM$. Let $\Omega$ the circumcircle of triangle $\triangle AMX$ a) Show that $CM$ is tangent to $\Omega$ b) Show that the lines $NX$ and $AC$ meet at $\Omega$