Incentre diagram of triangle ABC In triangle $ABC$, $AO$ and $BO$ bisect angles $BAC$ and \(

Question

Incentre diagram of triangle ABC

In triangle $ABC$, $AO$ and $BO$ bisect angles $BAC$ and $ABC$, respectively. $AO$ and $BO$ intersect at $O$.

Prove the concurrency (intersecting at one point) of the $3$ angle bisectors of a triangle by proving that $CO$ bisects angle $ACB$.
Hence prove that there exists a circle with centre $O$ that touches $AB$, $BC$ and $AC$ at one point, respectively.

Note: The point $O$ is known as the incentre of triangle $ABC$ and $O$ is equidistant to the $3$ sides of triangle $ABC$.

Accepted Answer

(i) $CO$ bisects $\angle ACB$, so the three internal bisectors concur at $O$. (ii) The circle centred at $O$, of radius $OD=OE=OF$, touches all three sides.

Concurrency of the Angle Bisectors of Triangles and the Incentre of Triangles

Concurrency of the Angle Bisectors of Triangles and the Incentre of Triangles

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