If the incircle of the triangle ABC passes through its circumcenter, then find the value of `4 sin.(A)/(2) sin.(B)/(2) sin.(C)/(2)`

Answer

`sqrt2 -1`

Explanation

Distance between circumcenter (O) and incenter (I) is `sqrt(R^(2) -2r R)`
If incircle of triangle ABC passes through it's circumcenter,
`sqrt(R^(2) - 2r R) = r`
`rArr ((r)/(R))^(2) + 2((r)/(R)) - 1 = 0`
`rArr (r)/(R) = (-2 +- sqrt(4 + 4))/(2)`
`rArr (r)/(R) = sqrt2 -1` (as `(r)/(R) gt 0`)
`rArr 1 + (r)/(R) = sqrt2`
`rArr 4 sin.(A)/(2) sin.(B)/(2) sin.(C)/(2) = sqrt2 -1`