in triangle ABC, the angular bisectors of angle B and angle C meet at I. from I, perpendiculars ID and IE are drawn to AB and AC respectively. prove that:
1) triangle ADI is congruent to triangle AEI
2) prove that AI bisects angle A
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Step-by-step explanation:
Given A △ABC in which the bisectors of ∠B and ∠C meet the sides AC and AB at D and E respectively.
To prove AB=AC
Construction Join DE
Proof In △ABC, BD is the bisector of ∠B.
∴
BC
AB
=
DC
AD
...........(i)
In △ABC, CE is the bisector of ∠C.
∴
BC
AC
=
BE
AE
.......(ii)
Now, DE∣∣BC
⇒
BE
AE
=
DC
AD
[By Thale's Theorem]......(iii)
From (iii), we find the RHS of (i) and (ii) are equal. Therefore, their LHS are also equal i.e.
BC
AB
=
BC
AC
⇒ AB=AC
Hence, △ABC is isosceles.
Sorry for typing error
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