DEF is the triangle formed by joning the points of contact of the incircle with the sides of the triangle ABC; prove that
(i)its sides are 2r cos 
, 2r cos 
and 2r cos 
,
(ii)its angles are 
– 
, 
– 
and 
– 
and
(iii) its area is 
, i.e. 
Δ .
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) E Ι FA is a cyclic quadrilateral
∴ 
= A Ι
A Ι = r cosec A/2
∴ EF = r cosec A/2.sin A
= 2 r cos A/2
similarly DF = 2 r cos B/2
and DE = 2r cos C/2.
(ii) Ι ECD is a cyclic quadrilateral
∴ ∠ Ι CE = ∠ Ι DE = 
similarly ∠ Ι DF = ∠ Ι BF = 
∴ ∠ FDE = 
= 
= 
– 
(iii) area of Δ DEF = 
FD . DE sin ∠ FDE
= 
= 2r 2 
= 2r 2 
= 
= 
= 
= 
= 
= 
= 
.
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