IMO 1979 LL FRA22

Origin: FRA

Problem

Consider two quadrilaterals ABCD and A′B′C′D′ in an affine Euclidian plane such that AB = A′B′, BC = B′C′, CD = C′D′, and DA = D′A′. Prove that the following two statements are true: (a) If the diagonals BD and AC are mutually perpendicular, then the diagonals B′D′ and A′C′ are also mutually perpendicular. (b) If the perpendicular bisector of BD intersects AC at M, and that of B′D′ intersects A′C′ at M ′, then MA MC = M′A′ M′C′ (if MC = 0 then M ′C′ = 0).