a/sinA=b/sinB=c/sinC=2R
sinA=a/(2R),sinB=b/(2R),sinC=c/(2R)
sin²A+sin²B+sin²C=2√3 sinAsinBsinC
[a/(2R)]^2+[b/(2R)]^2+[c/(2R)]^2=2√3a/(2R)*b/(2R)*c/(2R)
a^2+b^2+c^2=2√3*abc/(2R),又c^2=a^2+b^2-2abcosC
2a^2+2b^2-2abcosC=2√3*absinC
a^2+b^2=abcosC+√3*absinC
2cos(60度-C)=a/b+b/a
a/b+b/a>=2(根据不等式的性质)
2cos(60度-C)>=2
cos(60度-C)>=1
所以cos(60度-C)=1
C=60度
同理,A=60度,B=60度,三角形为等边三角形