Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be a point in the interior of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Let
![A'](/media/m/9/2/6/9267b8bcabe1ad2df0d51dab3364714b.png)
lie on
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
with
![MA'](/media/m/a/6/4/a64848157b72523c0f6f31982b38b2c5.png)
perpendicular to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. Define
![B'](/media/m/a/1/a/a1a88eb7f35fee4f41c66bfb0c902f51.png)
on
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
and
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
on
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
similarly. Define
Determine, with proof, the location of
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
such that
![p(M)](/media/m/2/7/f/27f8fff2ffd352e29397aea4f54d8d8e.png)
is maximal. Let
![\mu(ABC)](/media/m/4/1/a/41a1454bbd9671f4972e134bc977ffe4.png)
denote this maximum value. For which triangles
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is the value of
![\mu(ABC)](/media/m/4/1/a/41a1454bbd9671f4972e134bc977ffe4.png)
maximal?
%V0
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?