A segment, ray, line or palne that is perpemndicular to a segment at its midpoint is called a **perpendicular bisector**.

**Theorem** 1.1 *Perpendicular Bisector Theorem*

If a point is on perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

If CP is the perpendicular bisector of AB, then CA = CB

**Theorem** 1.2 Converse of the Per*pendicular Bisector Theorem*

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

If DA = DB, then D lies on the perpendicular bisector of AB.

**Definition**: The **distance from a point to a line** is define as the length of the perpendicular segment from the point to the line.

**Definition**: When a pint is the same distance from one line as it is from another line, then the point is equidistant from the two lines.

**Theorem** 2.1 *Angle Bisector Theorem*

If a point is on the bisector of an anlgle, then it is equidistant from the two sides of the angle.

If **∠**BAD = **∠**CAD, then DB = DC.

**Theorem** 2.2 *Converse of the Angle Bisector Theorem *

If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

If DB = DC, then **∠**BAD = **∠**CAD

* Definition: *When three or more lines intersect in the same point, they are called

*The point of intersection of lines is called the*

**concurrent lines.**

**point of concurrency.*** Definition:* The point of concurrency of the perpendicular bisectors of triangles is called the

**circumcenter of the triangle**.**Theorem** 3.1 Concurrency of Perpendicular Bisectors of a Triangle

The perpendicular bisector of a triangle intersect at a point that is equidistant from the vertices of the triangle.

PA = PB = PC

* Definition:* The point of concurrency of the angle bisectors is called the incentsr of the triangle and it always lies inside the triangle.

**Theorem** 4.1 Concurrency of Perpendicular Bisectors of a Triangle

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

PD = PE = PF

* Definition:* A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.

* Definition:* The point of concurrency of the medians is called the centroid of the triangle. The centroid is always inside the triangle.

**Theorem** 5.1 Concurrency of Medians of a Triangle

The medians of a triangle intresect at a point that is two third of distance from each vertex to the midpoint of the opposite side.

If P is the centroid of ΔABC, then * *

* AP= 2/3 AD , BP = 2/3 BF, CP = 2/3 CE*

* Definition:* An

**altitude**of a triangle is a perpendicular segment from vertex to the opposite side or the line that contains the opposite side.

* Definition: *The point of concurrency of the altitudes called the

**orthocenter**of the triangle.

**Theorem** 6.1 Concurrency of Altitudes of a Triangle

The lines containing the altitudes of a triangle are concurrent.

If AE, BF and CD are altitudes of ΔABC, then the lines AE, BF and CD intersect at some point H.