證明:
設a,b是兩個鄰邊,c,d是兩個鄰邊。a,b夾角v是一段圓弧的圓周角,其對應的弦長為x。c,d夾角是這段圓弧非的圓周角,等於180-v。
x^2=a^2+b^2-2ab cos(v)=c^2+d^2-2ab cos(180-v)
cos(v)=(a^2+b^2-c^2-d^2)/2(ab+cd) (1)
A=ab sin(v)/2 + cd sin(180-v)/2=(ab+cd) sin(v)/2
A^2=(ab+cd)^2sin^2(v)/4 (2)
(1)代入(2)得:
A^2=(ab+cd)^2(1-(a^2+b^2-c^2-d^2)^2/(2(ab+cd))^2)/4=
((2(ab+cd))^2-(a^2+b^2-c^2-d^2)^2)/16=
((2(ab+cd))+(a^2+b^2-c^2-d^2)((2(ab+cd))-(a^2+b^2-c^2-d^2))/16=
((a+b)^2-(c-d)^2)((c+d)^2-(a-b)^2)/16=
(a+b+c-d)(a+b-c+d)(c+d+a-b)(c+d-a+b)/16=
(a+b+c+d-2d)(a+b+c+d-2c)(c+d+a+b-2b)(c+d+a+b-2a)/16=
(s-a)(s-b)(s-c)(s-d)