On a line l there are three different points A, B and P in that order. Let a be the line through A perpendicular to l, and let b be the line through B perpendicular to l. A line through P, not coinciding with l, intersects a in Q and b in R. The line through A perpendicular to BQ intersects BQ in L and BR in T. The line through B perpendicular to AR intersects AR in K and AQ in S.
(a) Prove that P, T, S are collinear.
(b) Prove that P, K, L are collinear.(2nd Benelux Mathematical Olympiad 2010, Problem 3) geometrygeometry proposed