Question 115
The medians $AM_A$, $BM_B$, and $CM_C$ of a triangle $ABC$ meet at $M$. Construct a circle $\Omega_A$ passing through the midpoint of $AM$ and tangent to $BC$ at $M_A$. Construct the circles $\Omega_B$ and $\Omega_C$ similarly. Prove that the circles $\Omega_A$, $\Omega_B$, and $\Omega_C$ share a common point.