In a Triangle the Bisectors Intersect

The three bisectors of any triangle intersect.

Let ABC be a triangle and AA1, BB1 and CC1 be its bisectors. Then AA1, BB1 and CC1 intersect in a point I.


Proof:
A point is on the interior bisector line of an angle if and only if the distances to the two sides of the angle are equal.

Let us use this property to show that interior angle bisectors pass by one point.Consider $ I=BB_1\cap CC_1 $ As $ I\in BB_1 $ we have IH=IF. As $ I\in CC_1 $ we have IH=IJ and so IF=IJ and I belongs to the interior angle bisector of $ \angle BAC $

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