Prove that `("a c o s"A+b cosB+c cosC)/(a+b+c)=r/Rdot`

Explanation

We have
`a cos A + b cos B + c cos C`
`= R(2 sin A cos A + 2 sin B cos B + 2 sin A sin C)`
`= R (sin 2A + sin 2B + sin 2C)`
`= 4R sin A sin B sin C`
and `a + b + c = 2R (sin A + sin B + sin C)`
`= 8 R cos (A//2) cos (B//2) cos (C//2)`
`rArr (a cos A + b cos B + c cos C)/(a + b + c)`
`= (4R sin A sin B sin C)/(8R cos A //2 cos B//2 cos C//2)`
`= ((2 sin.(A)/(2) cos.(A)/(2))(2sin .(B)/(2) cos.(B)/(2)) (2sin.(C)/(2) cos.(C)/(2)))/(2cos.(A)/(2) cos.(B)/(2) cos.(C)/(2))`
`= 4 sin.(A)/(2)sin.(B)/(2) sin.(C)/(2) = (r)/(R)`