Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
Find and verify a coordinate circumcenter
Problem
For \(A(0,~0)\), \(B(6,~0)\), and \(C(0,~8)\), find the circumcenter from the perpendicular-bisector relationships. Verify that its squared distance to each vertex is the same, then state the radius and center-radius equation of the circle.
Coordinate circumcenter work has two jobs: locate a candidate center and verify that it is equidistant from every vertex. We’ll exploit the horizontal and vertical sides to write simple perpendicular bisectors, intersect them, compare squared distances, and only then write the center-radius equation.
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