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If I is the incenter of Delta ABC and R(...

If `I` is the incenter of `Delta ABC and R_(1), R_(2), and R_(3)` are, respectively, the radii of the circumcircle of the triangle IBC, ICA, and IAB, then prove that `R_(1) R_(2) R_(3) = 2r R^(2)`

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Let `I` be the incentre of `Delta ABC`
In `Delta IBC, angle BIC = pi - (B + C)/(2) = pi - (pi -A)/(2) = (pi + 4)/(2)`
Now, the radius of circumcircle of `Delta IBC`, by sine rule, is
`R_(1) = (BC)/(2 sin(angle BIC)) = (a)/(2 sin((pi + A)/(2)))`
`= (2R sin A)/(2 cos.(A)/(2))`
`= 2R sin.(A)/(2)`
Similarly, the radii of circumcircle of `Delta ICA and Delta IAB` are given by
`R_(2) = 2R sin. (B)/(2) and R_(3) = 2R sin. (C)/(2)`
`rArr R_(1) R_(2) R_(3) = 8R^(3) sin. (A)/(2) sin.(B)/(2) sin.(C)/(2) = 2rR^(2)`
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