**Geometry Chapter 5.2 Bisectors of a Triangle**

**I) Vocabulary:**

A) A Perpendicular bisector of a triangle- is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

A) A Perpendicular bisector of a triangle

**B) Concurrent lines**- when three or more lines (or rays or segments) intersect in the same point.

**C) Point of concurrency**- the point of intersection of the concurrent lines.

**D) Circumcenter of the triangle**- the point of concurrency of the perpendicular bisectors of a triangle.

**E) Angle bisector of a triangle**- is a bisector of an angle of the triangle.

**F) Incenter of the triangle**- the point of concurrency of the angle bisectors of a triangle.

**II) Theorems**:

**A) 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.

**Example**: Given ΔABC with point P being the circumcenter of the triangle, we can conclude that

PA = PB = PC.

Using point P as the center of the circle and PA, PB, or PC as a radius, the circleis circumscribed about the triangle.

**B) 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 triangle.

**Example**: Given ΔABC with point P being the incenter of the triangle, we can conclude that

PD = PE = PF and PD ll BC, PE ll AC and PF ll AB.

Using point P as the center of the circle and PD, PE, or PF as a radius, the circle is inscribed within the triangle.