Prove that if A and A' and B and B' are 2 pairs of points symmetric about a line XY, then the 4 points lie on the same circle.
I thought about it for a long time, until I obtained this proof:
Since A and A' and B and B' are symmetric about XY, A and A' are equidistant from XY and so are B and B'. Now, because XY is the locus of points equidistant from A, A' and B, B', there is bound to be at least one point, say R, which is equidistant from all points A, A', B, and B'. That is, AR = A'R = BR = B'R. $\therefore$, the points A, A', B, B' are the geometric locus of R, which is a circle. Thus A, A', B, B' are on the same circle.
After writing this proof, I felt a bit uncomfortable, because something didn't seem quite right with the way that I wrote it. If anyone could give suggestions or point out if I was wrong, that would be greatly appreciated.