cosC = (a^2+b^2-c^2)/2ab
S=1/2*ab*sinC=1/2*ab*√(1-cos^2 C)
=1/2*ab*√[1-(a^2+b^2-c^2)^2/4a^2*b^2]
=1/4*√[4a^2*b^2-(a^2+b^2-c^2)^2]
=1/4*√[(2ab+a^2+b^2-c^2)(2ab-a^2-b^2+c^2)]
=1/4*√[(a+b)^2-c^2][c^2-(a-b)^2]
=1/4*√[(a+b+c)(a+b-c)(a-b+c)(-a+b+c)]
设p=(a+b+c)/2
则p=(a+b+c)/2,p-a=(-a+b+c)/2,p-b=(a-b+c)/2,p-c=(a+b-c)/2,
上式=√[(a+b+c)(a+b-c)(a-b+c)(-a+b+c)/16]
=√[p(p-a)(p-b)(p-c)]