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.

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 t m n l P l, m and n are concurrent at P s, v, and t are NOT concurrent

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. A B C l m n

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. A B C l m n

The point of concurrency of the altitudes is called the orthocenter. 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. A B C F D E Note: The orthocenter is not always inside the triangle!

The point of concurrency of the medians is called the centroid. 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. C A B M N P The centroid of a triangle is important in physics because it is the center of gravity of the triangle!

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

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