InterviewSolution
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InterviewSolution
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Prove that `("a c o s"A+b cosB+ccosC)/(a+b+c)=r/Rdot` |
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Answer» 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)` |
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