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. 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

3. 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

4. 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

5. 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!

6. 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!

7. 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 = = 9

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