ISL 2001 G4

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

$$p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.$$

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?

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