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Let I be the incetre of Delta ABC having...

Let I be the incetre of `Delta ABC` having inradius r. Al, BI and Ci intersect incircle at D, E and F respectively. Prove that area of `DeltaDEF " is " (r^(2))/(2) (cos.(A)/(2) + cos.(B)/(2) + cos.(C)/(2))`

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`angleBIC =pi -(B+C)/(2) = pi -(pi -A)/(2) = (pi)/(2) + (A)/(2)`

Now, area of `DeltaEIF = (1)/(2) EI xx FI xx sin ((pi)/(2) + (A)/(2))`
`= (1)/(2) r^(2) cos.(A)/(2)`
`:.` Area of `DeltaDEF = DeltaEIF + DeltaDIF + DeltaDIE`
`=(r^(2))/(2) (cos.(A)/(2) + cos.(B)/(2) + cos.(C)/(2))`
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