Lesson 14 3 The Concurrence Theorems By the

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 m P t n l 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 m n B l C

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 l n B m C

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 Note: The orthocenter is not always inside the triangle! D F B E C

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

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

Summary How will you remember the difference between the circumcenter, incenter, orthocenter, and centroid? Homework: WS 14. 3