Question 152
The triangle $ABC$ has circumcentre $O$ and circumcircle $\Gamma$. Let $AI$ be a diameter of $\Gamma$. The ray $AI$ extends to intersect the circumcircle $\omega$ of $\angle BOC$ for the second time at a point $P$.
Let $AD$ and $IQ$ be perpendicular to $BC$, with $D$ and $Q$ on $BC$. Let $M$ be the midpoint of $BC$.
1.
Prove that $|AD| \cdot |QI| = |CD| \cdot |CQ| = |BD| \cdot |BQ|$.
2.
Prove that $IM$ is parallel to $PD$.
Let $AD$ and $IQ$ be perpendicular to $BC$, with $D$ and $Q$ on $BC$. Let $M$ be the midpoint of $BC$.
1.
Prove that $|AD| \cdot |QI| = |CD| \cdot |CQ| = |BD| \cdot |BQ|$.
2.
Prove that $IM$ is parallel to $PD$.