Lesson 14.3 The Concurrence Theorems

1. Lesson 14.3 The Concurrence Theorems By the end of this lesson you will be able to identify the circumcenter, the incenter, the orthocenter, and the centroid of a triangle.

2. s m v P t n l Lines that have exactly one point in common are said to be concurrent. …….thus we have……. Definition: Concurrent Lines are lines that intersect in a single point s, v, and t are NOT concurrent l, m and n are concurrent at P

3. A m n C B l Theorem The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices of a triangle. The point of concurrency of the perpendicular bisectors is called the circumcenter.

4. A l m n C B Theorem The bisectors of the angles of a triangle are concurrent at a point that is equidistant from the sides of a triangle. The point of concurrency of the angle bisectors is called the incenter.

5. A D F C B E Theorem The lines containing the altitudes of a triangle are concurrent at a point. The point of concurrency of the altitudes is called the orthocenter. Note: The orthocenter is not always inside the triangle!

6. A N P C B M Theorem The medians of a triangle are concurrent at a point that is 2/3 of the way from any vertex of the triangle to the midpoint of the opposite side. The point of concurrency of the medians is called the centroid. The centroid of a triangle isimportant in physics because it is the center of gravity of the triangle!

7. P T C R Q S Example: In triangle PQR, the medians, QT and PS are concurrent at C. PC = 4x – 6 CS = x Find: • x • PS • Answers: • x = 3 • PS = PC + CS = 6 + 3 = 9

8. Summary How will you remember the difference between the circumcenter, incenter, orthocenter, and centroid? Homework: WS 14.3