1. 5.2 Bisectors of a Triangle Advanced Geometry

2. PERPENDICULAR BISECTOR OF A TRIANGLE A perpendicular bisector of a triangle is a line (ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

3. 3 Perpendicular Bisectors in every triangle!

4. VOCAB Concurrent lines – Three or more lines (rays or segments) that intersect at the same point. Point of Concurrency – The point of intersection of concurrent lines. Circumcenter of the triangle – The point of concurrency of the perpendicular bisectors of a triangle.

5. Point of Concurrency Circumcenter of the Triangle The circumcenter of the triangle can be inside the triangle, on the triangle, or outside the triangle.

6. CONCURRENCY OF PERPENDICULAR BISECTORS OF A TRIANGLE The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

7. Where to meet?

8. Angle Bisector of a Triangle An angle bisector of a triangle is a bisector of an angle of the triangle.

9. INCENTER OF THE TRIANGLE The point of concurrency of the angle bisectors is called the incenter of the triangle.

10. Incenter always lies inside the triangle!

11. CONCURRENCY OF ANGLE BISECTORS OF A TRIANGLE The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangles

12. Example 1.What segments are congruent? 2.If AY = 15 and AP = 17, then find PY and PX.

13. How to remember points of concurrency? ABI loves to eat Peanut Butter Cups. Peanut Butter Cups: Perpendicular Bisectors – Circumcenter ABI: Angle Bisectors – Incenter

14. Remember! Circumcenter is equidistant to the verticies! – Can be inside, on, or outside the triangle Incenter is equidistant to the sides! – Is always inside the triangle