Chứng minh rằng: R/r = sin A + sin B + sin C/2sin Asin Bsin C
Chứng minh rằng: \[\frac{R}{r} = \frac{{\sin A + \sin B + \sin C}}{{2\sin A\sin B\sin C}}\].
Lời giải
Ta có: \(\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}} = 2R\)
\(S = \frac{{a + b + c}}{{2r}} = R.r.\left( {\sin A + \sin B + \sin C} \right)\)
Mặt khác: \(S = \frac{{abc}}{{4R}} = \frac{{2R\sin A.2R\sin B.2R\sin C}}{{4R}} = 2{R^2}\sin A\sin B\sin C\).
Nên \(R.r.\left( {\sin A + \sin B + \sin C} \right) = 2{R^2}\sin A\sin B\sin C\).
Hay \[\frac{R}{r} = \frac{{\sin A + \sin B + \sin C}}{{2\sin A\sin B\sin C}}\].