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prove that internal bisectors of the angles of a triangle are conurrent. also find the position voctor of the point of concurrency.

prove that internal bisectors of the angles of a triangle are conurrent. also find the position voctor of the point of concurrency.


 

Grade:12

1 Answers

Aman Bansal
592 Points
12 years ago

Dear Prashant,

Let the vertices of the triangle by A(
We thus have
(BD/DC) = c/b
Thus, D is given by (the internal division formula):
In ΔABD, since BI is the angle bisector, we have
Thus, we now have the position vectors of A and D we know what ratio I divides AD in. I can now be easily determined using the internal division formula:
The symmetrical nature of this expression proves that the bisector of C will also pass through I. The angle bisectors will therefore be concurrent at I, called the incentre. The position vector of the incentre is given by (1).

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