Proof that the perpendicular bisectors of the sides of a general triangle meet at a point

Question

I have been studying Lang's basic math and I am stuck on the problem below:

This is my representation:

I am not sure on how I should proceed in this proof, I think this page is related to the problem, if so should I use a similar approach in the exercise? Thanks in advance.

Answer 1

HINT: Points on the perpendicular bisector of a segment are equidistant from the endpoints of that segment (this is not difficult to show if you look at the right triangles formed by the bisector, the segment, and the distances to the endpoints). Then, in your figure, you have $OQ=OM$ and $OQ=OP$. Now you can easily complete the proof.