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.

Does MSE question 1628698 "General Formula for Equidistant Locus of Three Points" answer your question? — Somos, Jun 20 '20 at 19:20
Does this answer your question? Prove that the perpendicular bisectors of the sides of a triangle meet at a point — Somos, Jun 20 '20 at 19:24
Thanks for the links but both of these while correct seem to approach the problem from a different perspective, I was looking for a proof similar to the one in the page I posted to keep consistent with the book I was learning from. — Cresio, Jun 20 '20 at 19:27

score 0 · Accepted Answer · answered Jun 20 '20 at 19:31

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.

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

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