婆罗笈多公式是针对圆的内接四边形的面积
\[
\prod_i(s-a_i)=((a_1-a_2)^2-(a_3+a_4)^2)((a_3-a_4)^2-(a_1+a_2)^2)
\]
\[
=16\sin^2(\alpha_1/2+\alpha_2/2)\cos^2(\alpha_1/2+\alpha_2/2)(\sin^2(\alpha_1/2-\alpha_2/2)-\cos^2(\alpha_3/2-\alpha_4/2))^2
\]
\[
2\sin(\alpha_1+\alpha_2)(\sin^2(\alpha_1/2-\alpha_2/2)-\cos^2(\alpha_3/2-\alpha_4/2))
\]
\[
=-\sin(\alpha_1+\alpha_2)(\cos(\alpha_1-\alpha_2)+\cos(\alpha_3-\alpha_4))
\]
\[
2S=\sin(2\alpha_1)+\sin(2\alpha_2)+\sin(2\alpha_3)+\sin(2\alpha_4)
\]
\[
=2\sin(\alpha_1+\alpha_2)(\cos(\alpha_1-\alpha_2)+\cos(\alpha_3-\alpha_4))
\]
看来是正确。原来算错了一点。
※ 修改:·supproton 于 Jul 26 17:17:14 2024 修改本文·[FROM: 117.155.182.*]
※ 来源:·水木社区
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