ILL 1978 Problem 5

Prove that for any triangle $ABC$ there exists a point P in the plane of the triangle and three points $A' , B'$ , and $C'$ on the lines $BC, AC$ , and $AB$ respectively such that $$AB \cdot PC'= AC \cdot PB'= BC \cdot PA'= 0.3M^2,$$ where $M = max\{AB,AC,BC\}$ .