a/sinA=b/sinB=c/sinC
令a/sinA=b/sinB=c/sinC=1/k
则sinA=ak,sinB=bk,sinC=ck
带入sinA+sinB=√2sinC
ak+bk=√2ck
a+b=√2c
周长=a+b+c=√2c+c=c(√2+1)=√2+1
所以AB=c=1
c=1,a+b=√2c=√2
S=absinC/2=1/6sinC
所以ab=1/3
a+b=√2
两边平方
a^2+2ab+b^2=2
a^2+b^2=2-2ab=4/3
cosC=(a^2+b^2-c^2)/2ab=(4/3-1)/2*(1/3)=1/2
所以C=π/3