1. Objectives: To define points of concurrency in triangles
5.3 (day 2) Concurrency
Objectives: To define points of concurrency in triangles

2. Definitions
Concurrent: when three or more lines intersect in one point, they are said to be concurrent
Point of concurrency: the point at which the lines intersect
For any triangle, there are four different sets of concurrent lines.

3. Points of Concurrency
Incenter: the point of concurrency of the angle bisectors of a triangle
Circumcenter: the point of concurrency of the perpendicular bisectors of a triangle
Orthocenter: the point of concurrency of the altitudes of the triangle
Centroid: the point of concurrency of the medians of a triangle

4. Points of Concurrency

5. Concurrency Theorems
The circumcenter is equidistant from the vertices of the triangle.
The incenter is equidistant from the sides of the triangle.

6. Definitions
Inscribed: A circle is inscribed in a polygon if the sides of the polygon are tangent to the circle

7. Circumscribed: A circle is circumscribed about a polygon if the vertices of the polygon are on the circle.

8. Incenter of a Triangle
The incenter of a triangle is the center of the circle that is inscribed in it.

9. Circumcenter of a Triangle
The circumcenter of a triangle is the center of a circle that is circumscribed about it.

10. Example
Find the center of the circle circumscribed around the triangle with vertices (0, 0), (-8, 0), (0, 6).

11. Theorem
The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint.

12. Example If WM = 16, then WX = ____ If RZ = 30, then MZ = ____
If MY = 3, then MO = ____ W Z Y M R O X

13. Center of Gravity
The centroid of a triangle is the center of mass or also the center of gravity. A triangle could be balanced on its centroid.

14. Assignment
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