Math's: Concurrent Lines: Concept, Angle Bisector of a Triangle and Perpendicular Bisector of the Sides of a Triangle

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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

Concept of Concurrent Lines

Concept of Concurrent Lines

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.

Concurrent Lines

Concurrent Lines

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.

Angle Bisector of Triangle

Angle Bisector of 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.

Perpendicular Bisector

Perpendicular Bisector

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.

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