An applet to show that the three perpendicular bisectors of the sides of a triangle all meet in a single point.

The three vertices of the triangle above can each be dragged to show that the perpendicular bisectors of the three sides still meet in a single point when we look at other triangles
When a mathematician sees a behavior like this that works with all triangles (or at least with a bunch of examples that we have looked at) the suspicion is that there must be a structure that helps us prove that it must always happen.
The check boxes in the diagram provide the framework for the proof. Click the box labeled "Prelim Circle". From any point on a perpendicular bisector, we can construct a circle centered at that point, going through the two vertices that form the end of the segment. Moving down the bisector we reach a point where the circle goes through all three vertices. That point must be on all three perpendicular bisectors. We can see that circle directly with the box marked "Final Circle".