Method to find the common meeting point of three lines lying on a unique line with other constraints .

Question

Assume that perpendicular distances from $P,Q,R,S,T,U$ to the given line XY is known to us and also that $P,Q,R$ are fixed points and the XY line too fixed and also $ST=TU$ . What should be the locations of $S,T,U$ (in that order) be such that the line segments formed by joining $PS , QT, RU$ meet at a common point which will lie on line $XY$ ?. Progress : considering the mirror reflection of all the three points $S,T,U$ along $XY$ now we want that all three points $P,Q,R$ rays reflect from a common point on $XY$ to meet at $S',T',U'$ in the other side , does this help ? If not what should be the way to do so ?Note: $P,Q,R$ lie along a fixed line , similarily $S,T,U$ forms a straight path.