# Math՚s: Concurrent Lines: Concept, Angle Bisector of a Triangle and Perpendicular Bisector of the Sides of a Triangle (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Concept of Concurrent Lines

**Three lines in a plane may**:

- be parallel to each other, i.e.. , intersect in no point or
- intersect each other in exactly one point or
- intersect each other in two points or
- intersect each other at the most in three points

Three or more lines in a plane which intersect each other in exactly one point, or which pass through the same point are called concurrent lines and the common point is called the point of concurrency.

## Angle Bisector of a Triangle

A line which divides an angle of a triangle into two equal parts is known as an angle bisector of the triangle. Since a triangle has three angles, we can draw three angle bisectors in it. are the three angle bisectors of given below.

- All the three angle bisectors of a triangle pass through the same point, that is they are concurrent
- The point of concurrency is called the ‘Incentre’ of the triangle.
- Taking as the centre we can draw a circle touching all the three sides of the triangle called ‘Incircle’ of the triangle.

## Perpendicular Bisector of the Sides of a Triangle

A line which bisects a side of a triangle at right angle is called the perpendicular bisector of the side. Since a triangle has three sides, so we can draw three perpendicular bisectors in a triangle. are the three perpendicular bisectors of given below.

- The three perpendicular bisectors of the sides of a triangle always pass through the same point, that is, they are concurrent.
- The point of concurrency is called the ‘circumcenter’ of the triangle.
- If we take O as the center and AO as the radius, we can draw a circle passing through the three vertices, A, B and C of the triangle, called ‘Circumcircle’ of the triangle.

It is to be noted here that the circumcenter will be

- In the interior of the triangle for an acute triangle
- On the hypotenuse for a right triangle and
- In the exterior of the triangle for an obtuse triangle.