Special Segments of Triangles

1. Special Segments of Triangles Sections 5.2, 5.3, 5.4

2. Perpendicular bisector theorem • A point is on the perpendicular bisector if and only if it is equidistant from the • endpoints of the segment.

3. Angle Bisector Theorem • A point is on the bisector of an angle if and only if it is equidistant from the two sides of the angle.

4. Medians of a triangle • A median of a triangle is a segment from a vertex to the midpoint of the • opposite side.

5. Altitudes of a triangle • An altitude of a triangle is the perpendicular segment from a vertex to the • opposite side or to the line that contains the opposite side.

6. Concurrency • The point of intersection of the lines, rays, or segments is called the point of concurrency.

7. Points of concurrency • The point of concurrency of the three perpendicular bisectors a triangle is called the circumcenter. • The point of concurrency of the three angle bisectors of a triangle is called the incenter. • The point of concurrency of the three medians of a triangle is called the centroid. • The point of concurrency of the three altitudes of a triangle is called the orthocenter. • The incenter and centroid will always be inside the triangle. The circumcenter • and orthocenter can be inside, on, or outside the triangle.

8. What is special about the Circumcenter? • The perpendicular bisectors of a triangle intersect at a point that is equisdistant • from the vertices of the triangle. PA = PB = PC

9. What is special about the Incenter? • The angle bisectors of a triangle intersect at a point that is equidistant from the • sides of the triangle. PD = PE = PF

10. What is special about the Centroid? • The medians of a triangle intersect at a point that is two thirds of the distance • from each vertex to the midpoint of the opposite side.

11. What is special about the Orthocenter? • There is nothing special about the point of concurrency of the altitudes of a • triangle.