Two points A and B are on different sides of a line. Find a point Y on the line such that the absolute value of the difference from Y to A and Y to B is maximal.
My thoughts are as follows. Let's say that point Y is on a line but is not collinear with A and B. Then I could draw a triangle by joining the 3 points. For this triangle, segment AB is less than the sum of AY and BY. Then AB is greater than the absolute value of (YA-YB). How can I prove that this difference is maximal?