IMO 1985 LL GBR30

Origin: GBR

Problem

A plane rectangular grid is given and a “rational point” is defined as a point (x, y) where x and y are both rational numbers. Let A, B, A′, B′ be four distinct rational points. Let P be a point such that A′B′ AB = B′P BC = PA′ PA . In other words, the triangles ABP, A′B′P are directly or oppositely similar. Prove that P is in general a rational point and find the exceptional positions of A′ and B′ relative to A and B such that there exists a P that is not a rational point.