Let the incircle with center I of ` A B C` touch sides BC, CA and AB at D, E, F, respectively. Let a circle is drawn touching ID, IF and incircle of ` A B C` having radius `r_2dot` similarly `r_1a n dr_3` are defined. Prove that `(r_1)/(r-r_1)dot(r_2)/(r-r_2)dot(r_3)/(r-r_3)=(a+b+c)/(8R)`

Explanation

From the figure, in `Delta IQP`
`cos.(B)/(2) = (PQ)/(IP)`
Since `PQ = r_(2)`
and `IP = r - r_(2)` (as two circles touching internally), we have
`cos.(B)/(2) = (r_(2))/(r-r_(2))`
Similarly, for other such circles, `cos.(A)/(2) = (r_(1))/(r -r_(1)) and cos.(C)/(2) = (r_(3))/(r -r_(3))`
`(r_(1))/(r -r_(1)) .(r_(2))/(r -r_(2)) .(r_(3))/(r -r_(3)) = cos.(A)/(2) cos.(B)/(2) cos.(C)/(2)`
`= (1)/(4) (sin A + sin B + sin C)`
`= (a + b + c)/(8R)`