Question 100
Let $M, N$ and $P$ be the midpoints of the sides $BC, CA$ and $AB$, respectively, of the acute triangle $ABC$. Denote by $A'$, $B'$ and $C'$ the points diametrically opposite to the vertices $A, B$ and $C$, respectively, on the circumcircle of triangle $ABC$. On the open segments $MA'$, $NB'$ and $PC'$ consider the points $X, Y$ and $Z$, respectively, such that $\frac{MX}{XA'}=\frac{NY}{YB'}=\frac{PZ}{ZC'}$.
a) Prove that the lines $AX, BY$ and $CZ$ are concurrent at a point $S$.
b) Show that $OS \lt OG$, where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC$.
a) Prove that the lines $AX, BY$ and $CZ$ are concurrent at a point $S$.
b) Show that $OS \lt OG$, where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC$.