Question 18
Let M be a point in the interior of triangle ABC. Let $ A' $ lie on BC with $ MA' $ perpendicular to BC. Define $ B' $ on CA and $ C' $ on AB similarly. Define
$$ p(M)=\frac{MA^{\prime}\cdot MB^{\prime}\cdot MC^{\prime}}{MA\cdot MB\cdot MC}. $$
Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?
$$ p(M)=\frac{MA^{\prime}\cdot MB^{\prime}\cdot MC^{\prime}}{MA\cdot MB\cdot MC}. $$
Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?