若a,b,c是△ABC的三边,且acosA+bcosB=ccosC,判断△ABC的形状.

若a,b,c是△ABC的三边,且acosA+bcosB=ccosC,判断△ABC的形状.
若a,b,c是△ABC的三边,且acosA+bcosB=ccosC,判断△ABC的形状.

若a,b,c是△ABC的三边,且acosA+bcosB=ccosC,判断△ABC的形状.
直角三角形
a/sinA=b/sinB=c/sinC=t
a=tsinA
b=tsinB
c=tsinC
acosA+bcosB=ccosC
tsinAcosA+tsinBcosB=tsinCcosC
sin2A+sin2B=2sinCcosC
2sin(A+B)cos(A-B)=2sinCcosC
cos(A-B)=cosC
A-B=C
A=B+C－－－－－－＞A＝90°对应边为a
cosA=(b^2+c^2-a^2)/2bc (1)
cosB=(a^2+c^2-b^2)/2ac (2)
cosC=(b^2+a^2-c^2)/2ba (3)
(1)(2)(3)代入acosA+bcosB=ccosC得:a^2(b^2+c^2-a^2)+b^2(a^2+c^2-b^2)=c^2(b^2+a^2-c^2)
整理得:(a^2-b^2)^2=c^4
即:a^2-b^2=c^2或者b^2-a^2=c^2