Question 39
(1) Let $ABC$ be an acute triangle with $AB\lt AC$, and let $\Omega$ be its circumcircle. Let $B_0,C_0$ be the midpoints of $AC,AB$, and let $\Omega_1$ be the circumcircle of $AB_0C_0$. A circle $\omega$ passes through $B_0,C_0$ and is tangent to $\Omega$ at a point $X\ne A$. Let $a$ be the common tangent of $\Omega,\Omega_1$, and $x$ the common tangent of $\Omega,\omega$. Prove that $a,x,B_0C_0$ are concurrent.
(2) Let $D$ be the foot of the perpendicular from $A$ to $BC$, and $G$ the centroid of $ABC$. Prove that $D,G,X$ are collinear.
(2) Let $D$ be the foot of the perpendicular from $A$ to $BC$, and $G$ the centroid of $ABC$. Prove that $D,G,X$ are collinear.