Angle bisectors of triangle are concurrent

In this chapter we will attempt to prove if the angle bisectors of triangle are concurrent.

To understand the chapter, you should have basic knowledge of triangle and congruency theorem.

Before moving on towards the main content, let us revise the basics first.

Basics of angle bisectors

What are angle bisectors ?

The line segment which divide the angle into two equal parts are called angle bisector.

Basics of angle bisector

In the above triangle ABC, line BO is the angle bisector since it divides ∠B into two equal halves.

How many angle bisectors in a triangle ?

Since there are three angles in a triangle, there can only be three angle bisectors in the triangle.

What do you understand by term ” Concurrent ” ?

In geometry, concurrent means that the line intersect at common point.

meaning of concurrent in geometry

In the above figure, line A, B, C & D are concurrent since they intersect at common point O.

Proving angle bisectors of triangle are concurrent

Consider the below triangle ABC in which line OA and OB are the angle bisectors.

To prove that angle bisectors are concurrent, we have to prove that the third line (OC) intersecting the point o is also an angle bisector line.

prove that angle bisectors of triangle are concurrent

ABC is a triangle.
OA & OB are the angle bisector lines.
∠OBC = ∠ OBA
∠OAB = ∠ OAC

To prove:
∠OCB = ∠ OCA

Draw perpendicular lines OM, ON and OP

Consider ▵OPB and ▵OMB

OB = BO { common side }
∠OBP = ∠ OBM { given }
∠OPB = ∠ OMB { 90 degree }

By AAS congruency, both the triangles are congruent.
i.e. \mathtt{\triangle OPB\ \cong \triangle OMB}

Since both triangle are congruent we can write OM = OP.

Similarly consider ▵OAP and ▵OAN

Again by AAS congruency we can say OP = ON

Similarly consider ▵OCM and ▵OCN.

Again by AAS congruency, we can write OM = ON.

On combining all the expressions we get OM = ON = OP.

Now consider triangle OCM and OCN.

OC = CO { common side }
OM = ON { proved above }
∠OMC = ∠ ONC { 90 degree }

By RHS congruency we can say that both triangles are congruent.
i.e. \mathtt{\triangle OCM\ \cong \triangle OCN}

Since both triangles are congruent, we can say that ∠OCM = ∠ OCN.

This means that ∠C is bisected by segment OC and OC is the angle bisector.

Hence we proved that OC is the angle bisector and all the angle bisector intersect at common point O

Next chapter : Prove that side opposite to equal angles are equal

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