由正弦定理知,a/sinA=b/sinB=c/sinC
∵acosA+bcosB=ccosC
∴sinAcosA+sinBcosB=sinCcosC
∴sin2A+sin2B=sin2C=sin(2π-2A-2B)=-sin(2A+2B)
∴0=sin2A+sin2B+sin(2A+2B)
=sin2A+sin2B+sin2Acos2B+sin2Bcos2A
=sin2A(1+cos2B)+sin2B(1+cos2A)
=4sinAcosAcos²B+4sinBcosBcos²A
=4cosAcosBsin(A+B)
∵sin(A+B)=sin(π-C)=sinC>0
∴cosA=0或cosB=0
∴A=π/2或B=π/2
∴△ABC是以a或b为斜边的直角三角形