Question

a+b+c/sina+sinb+sinc=a/sina

为什么a+b+c/sina+sinb+sinc=a/sina a+b-c/sina+sinb-sinc=a/sina等等请给予证明,同学说好像要用到初中知识,帮帮我吧

Answer 1

题目原意是证明"(a+b+c)/(sinA+sinB+sinC)=a/sinA"吧?很明显,对正弦定理用等比定理变换马上可得所证式,也可这样证:a=2RsinA,b=2RsinB,c=2RsinC(R为三角形ABC外接圆半径),故(a+b+c)/(sinA+sinB+sinC)=2R(sinA+sinB+sinC)/(sinA+sinB+sinC)=2R.且a/sinA=2R.故原式得证。