Theorems on concurrence of lines, segments, or circles associated with triangles all deal with three or more objects passing through the same point. Concurrence theorems are fundamental and proofs of them should be part of secondary school geometry.
1. Prove: The perpendicular bisector of the sides of a triangle meet at a point which is equally distant from the vertices of the triangle.
2. Construction: The point of concurrence of the perpendicular bisectors of the sides of a triangle is the circumcenter of the triangle and the circle through the three vertices with center at the circumcenter is the circumcircle. Show a construction for the circumcenter and circumcircle of any triangle.
3. Prove. The bisectors of the angles of a triangle meet at a point that is equally distant from the sides of the triangle.
4. The point of concurrence of the angle bisectors of a triangle is the incenter of the triangle and the circle with center at the incenter and tangent to each of the three sides of the triangle is the incircle. Show a construction for the incenter ane incircle of any triangle.
5. Prove. Let the sides of a triangle be extended so as to indicate the external angles. The bisectors of the external angles on one side of a triangle and the bisector of the opposite internal angle are concurrent.
6. The point of concurrency of the two external angle bisectors and the corresponding opposite internal angle bisector is an excenter of the triangle. The circle with center at the excenter and tangent to the lines of the sides (extended) of the triangle is an excircle. Each triangle has three excenters and three excircles. Show a construction for the excenters and excircles of a triangle.
7. Prove. The medians of a triangle are concurrent and intersect each other in a ratio of 2:1. The point of concurrency is called the centroid of the triangle.
8. Prove. The perpendiculars from each vertex to the line of the opposite side of a triangle are concurrent.
The point of concurrence of these perpendicular lines through each vertex is the orthocenter of the triangle.