Use coordinates to prove simple geometric theorems, including simple circle theorems.
Verify a diameter/right-angle claim with squared lengths
Problem
A circle has diameter endpoints \(A\)\((-5,0)\) and \(B\)\((5,0)\), with center \(O\) at their midpoint. Does \(C\)\((0,5)\) lie on the circle and make \(\angle~ACB\) a right angle? Use squared lengths to decide.
The diameter-angle theorem needs the test point to be on the circle first. We’ll derive the center and radius squared from the diameter endpoints, verify incidence by squared distance, compute all three triangle side squares, and use the converse Pythagorean relation to test the claimed angle independently.
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