Bisectors in Triangles Concurrency of Perpendicular Bisector Theorem If the perpendicular bisectors PX, PY and PZ are concurrent at P, then PA = PC = PB The point P is called the circumcenter of the triangle.

Proof P is on n, perpendicular to AB, so PA = PB P is on m, perpendicular to BC, so PB = PC Hence PA = PB = PC

The circle is circumscribed about the triangle.

The circumcenter may not always be inside the triangle.

Concurrency of Angle Bisector Theorem If the angle bisectors PA, PB and PC are concurrent at P, then PX = PY = PZ The point P is called the incenter of the triangle.

The circle is inscribed in the triangle.