Question 16
Given point $O$ inside the acute-angled triangle $ABC$, and point $O'$ inside the acute-angled triangle $A'B'C'$. $D$, $E$, $F$ are the feet of the perpendiculars from $O$ to $BC$, $CA$, $AB$ respectively, and $D'$, $E'$, $F'$ are the feet of the perpendiculars from $O'$ to $B'C'$, $C'A'$, $A'B'$ respectively. $OD$ is parallel to $O'A'$, $OE$ is parallel to $O'B'$ and $OF$ is parallel to $O'C'$. Also $OD - O'A' = OE - O'B' = OF - O'C'$. Prove that $O'D'$ is parallel to $OA$, $O'E'$ to $OB$ and $O'F'$ to $OC$, and that $O'D' \cdot OA = O'E' \cdot OB = O'F' \cdot OC$.