1. Objectives: Discover points of concurrency in triangles. Draw the inscribed and circumscribed circles of triangles. Warm-Up: How many times can you subtract the number 5 from 25? Once. After the first calculation you will be subtracting 5 from 20, then 5 from 15, and so on. If you divide thirty by a half and add ten, what is the answer? 70. When dividing by fractions, you must invert and multiply.

2. Vocabulary: Inscribed Circle: A circle in the inside of a triangle that touches each side at one point. (a circle is inscribed in a polygon if each side of the polygon is tangent to the circle)

3. Vocabulary: Circumscribed Circle: A circle that is drawn around the outside of a triangle and contains all three vertices. (a circle is circumscribed about a polygon if each vertex of the polygon lies on the circle)

4. Vocabulary: Concurrent: Literally “running together” of three or more lines intersecting at a single point.

5. Vocabulary: Incenter: The center of a inscribed circle; the point where the three angle bisectors intersect. (It is equidistant from the three sides of the triangle).

6. Vocabulary: Circumcenter: The center of a circumscribed circle where the three perpendicular bisectors of the sides of a triangle intersect. (It is equidistant from the three vertices of the triangle).

7. Example 1: Label the inscribed circle, circumscribed circle, the incenter, the circumcenter, and points of concurrency in the following figures. inscribed circle circumscribed circle incenter circumcenter points of concurrency

8. Example 2: Z Y X

9. Example 3: L M N

10. Example 4: J K L To find identify the perpendicular bisectors of the triangles sides.

11. Example 5: M N O To find identify the angle bisectors of the triangle.

12. Example 6: A BC To find identify the perpendicular bisectors of the triangles sides.

13. Example 7: W X Y To find identify the angle bisectors of the triangle.