Let
be a point in the interior of triangle
. Let
lie on
with
perpendicular to
. Define
on
and
on
similarly. Define
Determine, with proof, the location of
such that
is maximal. Let
denote this maximum value. For which triangles
is the value of
maximal?
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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?
Školjka