Suppose that A, B, X, and Y are collinear and distinct. If AX/AY = BX/BY, show that exactly one of the points

Question

A and B lies on line segment XY.

Answer 1

Let coordinates of A,B,X,Y on the line be denoted by a,b,x,y. Then the given equality can be written as
|x-a|/|y-a| = |x-b|/|y-b|
Without loss of generality assume that x
In the cases where A and B are not on the segment XY, a (x-a)/(y-a) = (x-b)/(y-b)
which leads to
(b-a)(y-x) = 0
Since y not= x (X and Y are distinct), a=b and so A and B are not distinct. Therefore these cases are not allowed.

In the remaining two cases where only B lies on XY, a (x-a)/(y-a) = -(x-b)/(y-b)
In the first case a Q.E.D.