The following exercise is from the book What is Mathematics by Richard Courant and Herbert Robbins:
Given two lines L, M and two points P, Q situated inside the angle formed by the two lines, the path of minimum length from P to L, then to M, and then to Q can be found as follows:
Let Q' be the reflection of Q in M and Q'' the reflection of Q' in L; draw PQ'' intersecting L in R and RQ' intersecting M in S; then R and S are the required points such that PR+RS+SQ is the path of minimum length from P to L to M to Q.
One might ask for the shortest path first from P to M, then to L, and from there to Q. In this case Q' is the reflection of Q in L and Q'' the reflection of Q' in M. R is the intersection of PQ'' with M and S the intersection of RQ' with L.
Show that the first path is smaller than the second if O (where O is the point of intersection between L and M) and R lie on the same side of the line PQ.
How do you do this?