OME 1990 Problem 5

On the sides $BC,CA$ and $AB$ of a triangle $ABC$ of area $S$ are taken points $A' ,B' ,C'$ respectively such that $AC' /AB = BA' /BC = CB' /CA = p$ , where $0 < p < 1$ is variable. (a) Find the area of triangle $A' B' C'$ in terms of $p$ . (b) Find the value of $p$ which minimizes this area. (c) Find the locus of the intersection point $P$ of the lines through $A'$ and $C'$ parallel to $AB$ and $AC$ respectively.