三角形已知3边求面积,三角形面积公式大全

解：x=a-y

h^2=b^2-y^2

h^2=c^2-x^2

x y=a

h^2=b^2-y^2=c^2-x^2

x^2-y^2=c^2-b^2

(x y)(x-y)=a(x-y)=c^2-b^2

x y=a

x-y=(c^2-b^2)/a

x=[a (c^2-b^2)/a]/2

=(a^2 c^2-b^2)/2a

y=[a-(c^2-b^2)/a]/2

=(a^2-c^2 b^2)/2a

x=(a^2 c^2-b^2)/2a

y=(a^2-c^2 b^2)/2a

h^2=b^2-y^2

=b^2-[(a^2-c^2 b^2)/2a]^2=

[4a^2b^2-(a^2-c^2 b^2)^2]/4a^2

S△ABC=ah/2

(S△ABC)^2=(ah/2)^2=a^2h^2/4

=[4a^2b^2-(a^2-c^2 b^2)^2]/16

=[(2ab)^2-(a^2-c^2 b^2)^2]/16

=(2ab a^2-c^2 b^2)(2ab-a^2 c^2-b^2)/16

=[(a b)^2-c^2][c^2-(a-b)^2]/16=

(a b c)(a b-c)(c a-b)(c-a b)/16

(S△ABC)^2=

(a b c)(a b-c)(c a-b)(c-a b)/16

令p=(a b c)/2

p-a=(a b c)/2-a=(b c-a)/2

p-b=(a b c)/2-b=(a c-b)/2

p-c=(a b c)/2-c=(a b-c)/2