`O` is the circumcenter of ` A B Ca n dR_1, R_2, R_3` are respectively, the radii of the circumcircles of the triangle `O B C ,O C A` and OAB. Prove that `a/(R_1)+b/(R_2)+c/(R_3) =(a b c)/(R_3)`

Explanation

If O is the circumcenter of `Delta ABC`, then
OA = OB = OC = R
Given that `R_(1), R_(2), and R_(3)` be the circumradii of `DeltaOBC, DeltaOCA and DeltaOAB`, respectively.
In `DeltaOBC`, using sine rule, `2R_(1) = (a)/(sin 2A) " or " (a)/(R_(1)) = 2 sin 2A`

Similarly, `(b)/(R^(2)) = 2 sin 2B and (c)/(R_(3)) = 2 sin 2C`
`rArr (a)(R_(1)) + (b)/(R_(2)) + (c)/(R_(3)) = 2 (sin 2A + sin 2B + sin 2C)`
`= 8 sin A sin B sin C`
`= 8(a)/(2R) (b)/(2R) (c)/(2R)`
`= (abc)/(R^(3))`