Triangle center/concurrency theorems

Circumcenter/perpendicular bisectors

• Thinking about circumscribed circles: the strategy
• Every point that is equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment
• Every point that is on the perpendicular bisector of a segment is equidistant from its endpoints
• Circumscribed circles: the proof. The perpendicular bisectors of a triangle are concurrent at the circumcenter of the triangle.

Incenter/angle bisectors

• Thinking about inscribed circles: the strategy
• Every point that is equidistant from the sides of an angle lies on the bisector of the angle (for angles smaller than a straight angle)
• Every point that is on the bisector of an angle is equidistant from its sides
• Inscribed circles: the proof. The angle bisectors of a triangle are concurrent at the incenter of the triangle.

Excenter (angle bisectors)

• Understanding the theorem: The angle bisector of the interior angle at vertex A of a triangle is concurrent with the two angle bisectors of the exterior angles at the other two vertices (where one side of each exterior angle is a subset of one of the rays of the interior angle at A); and the point of concurrency is the center of a circle which is tangent to the sides of the interior angle at A, and the opposite side of the triangle and which is exterior to the triangle
• Proving the theorem.