Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
Construct a circumcircle with perpendicular bisectors
Problem
Construct the circumcircle of acute triangle \(ABC\) using sides \(AB\) and \(AC\). State the \(\text{two}~\text{construction}~\text{lines}\) and their intersection \(O\), justify \(OA~=~OB~=~OC\), use \(\text{one}~\text{of}~\text{those}~\text{lengths}\) as the radius, and locate \(O\) relative to the triangle.
A circumcircle starts with equal distances to the vertices, so the useful loci are side perpendicular bisectors rather than angle bisectors. We’ll construct two independent loci, justify their intersection through equal-distance relationships, and then use that shared distance to complete and locate the circle.
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