Proofs with two circles: prove XY is perp. bisector of PQ

[tristanpilcher]

Two circles with centers X and Y intersect at points P and Q. Prove that XY is the perpendicular bisector of PQ.

My assignment is to prove the above statement in as many different ways as possible... using analytic methods, vector methods, converse statments, congruent triangles, similar triangles... anything that works. The circles in the proof should be two different sizes, but if I am really stuck, I can make them the same size.

I have tried vector proofs using the dot product. I could not find the proper way to start because there are so many ways to expand the dot product. I only ended up proving my opening statement that I was using to prove that XY was the perpendicular bisector of PQ.

I have also tried analytic methods, but I could not easily find the intersection points of the two circles.

Please help. I am not strong in proofs when it comes to circles. I do not know enough properties of circles to be able to use vector methods and I am confused as to where to start this problem. My teacher wants me to find the hardest methods to prove it as possible.

 

[galactus]

One of various methods would be to take a look at the diagram.

See the triangles?. Can you use them to prove what you need to prove?.

As for the vector equation of a circle:

If (x,y) is a point on the circle and (h,k) is the center, then x is the vector OB and c=OA. So, OB-OA=AB

vector equation of circle is ||x-c||=r