Theorem 1

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Incentre

Definition:

"The point of concurrency of the angle bisectors of the angles of triangle, is called the incentre". It is abbreviated as 'I'.

To determine the incentre of a triangle, it is just sufficient to find the point of intersection of its two angles. The third angle bisector is bound to pass through it by virtue of the below theorem.

Theorem

The angle bisectors of a triangle pass through the same point.

Note:

In the above theorem we have proved ID = IE = IF. That is, if a circle with centre I and radius = ID (or IE or IF) is drawn, the circle will pass through the points D, E and F. This circle is called the incircle of a triangle, the centre of incircle is called incentre and the radius of incircle is called inradius.