Let I be the incetre of ΔABC having inradius r. Al, BI and Ci intersect incircle at D, E and F respectively. Prove that area of ${\displaystyle \Delta DEF\text{is}\frac{r^2}{2}(\cos .\frac{A}{2}+\cos .\frac{B}{2}+\cos .\frac{C}{2})}DeltaDEF\; "\; is\; "\; (r^(2))/(2)\; (cos.(A)/(2)\; +\; cos.(B)/(2)\; +\; cos.(C)/(2))$

Explanation

`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))`