Question 27
Let $ABC$ be a triangle and let points $D, E \in (BC)$, $F, G \in (CA)$, $H, I \in (AB)$ such that $BD = CE$, $CF = AG$ and $AH = BI$. Consider $M, N, P$ the midpoint of the segments $GH$, $DI$, $EF$ respectively and let $M'$ be the intersection point of the lines $AM$ and $BC$.
a. Show that $\frac{BM'}{CM'} = \frac{AG}{AH} \cdot \frac{AB}{AC}$.
b. Prove that lines $AM$, $BN$ and $CP$ are concurrent.
a. Show that $\frac{BM'}{CM'} = \frac{AG}{AH} \cdot \frac{AB}{AC}$.
b. Prove that lines $AM$, $BN$ and $CP$ are concurrent.