Question

已知三角形ABC中角A=60a=根号3则a+b+c/sinA+sinB+sinC=

已知三角形ABC中角A=60a=根号3则a+b+c/sinA+sinB+sinC=

Answer 1

△ABC中a/sinA=b/sinB=c/sinC
(a+b+c)/(sinA+sinB+sinC)=a/sinA=根号3/sin60°=2

追问

(a+b+c)/(sinA+sinB+sinC)=a/sinA为啥相等

追答

设a/sinA=b/sinB=c/sinC=k
则a=ksinA，b=ksinB，c=sinC，代回即可得到

Answer 2

a/sinA=√3/(√3/2)=2
∴a/sinA=2R=2
∵a=2RsinA
b=2RsinB
c=2RsinC
∴(a+b+c)/(sinA+sinB+sinC)
=(2RsinA+2RsinB+2RsinC)/(sinA+sinB+sinC)
=2R(sinA+sinB+sinC))/(sinA+sinB+sinC)
=2R
=2