Write out a complete proof of the following:

THEOREM: The perpendicular bisectors of the sides of a triangle, if 2 of them meet, are concurrent at a point that is the center of a circle (circumcircle) passing through the vertices. (The point of concurrency is called the circumcenter of the triangle.)

NOTE:The conditional is needed here because in absolute geometry it is possible for 2 such perpendicular bisectors to be non-intersecting lines.

My work:

A picture I found to illustrate the theorem:

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(Side note: I don't understand what they mean by "the conditional is needed here...." Let me add that my professor is very confusing, and I am quite lost in this class. I read the book, but it doesn't help.

I don't even know where to begin. So what exactly am I being asked to prove? Is it to prove that the point O is the center of a circle? I know that = , = , and = by definition of perpendicular bisector.

But then, I can't quite see how knowing those will help me. As I stated in my previous post, I am unsure as to exactly what tools we can/can't use to proving statements because we are approaching geometry in a very different method than how we learned in high school. We are to basically "unlearn" what we were taught in high school, and build our proofing tools. I am just utterly lost.

Any help at all is greatly appreciated. Thank you very much for your time.