Let $ABC$ be a right angle triangle with $BC = AC =1$. let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR$, $APQ$ and $PQCR$. Find the minimum possible value of $M$.
Hint is appreciated.
Put your triangle on the usual Cartesian coordinate axes so that the points are $A(0,1),B(1,0),C(0,0)$.
Then let point $P$ be defined as $P(x,y)$.
Because $P$ lies on the line $AC$ you can express $y$ in terms of $x$.
Then you can find areas in terms of $x$ for easier comparison.
