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[几何] 反射三角形为退化的条件

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hbghlyj posted 2023-3-16 17:24 |Read mode
Last edited by hbghlyj 2023-3-17 00:13若\[ \cos A\cos B\cos C=-3/8 \]则$\triangle ABC$的反射三角形为退化的.
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original poster hbghlyj posted 2023-3-16 17:44
  1. In[]:= ASATriangle @@ (N[{a, 1, b}] /.
  2.    FindInstance[{a + b + c == Pi, 0 < a < b < c,
  3.       Cos[a] Cos[b] Cos[c] == -3/8, c - a > 1}, {a, b, c}][[1]])
  4. Out[]= Triangle[{{0, 0}, {1., 0}, {0.906983, 0.105422}}]
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用Asymptote画出

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original poster hbghlyj posted 2023-3-16 20:36
$A^*B^*C^*$共线等价于\[0=\left|
\begin{array}{ccc}
-1 & 2 \cos C & 2 \cos B \\
2 \cos C & -1 & 2 \cos A \\
2 \cos B & 2 \cos A & -1 \\
\end{array}
\right|=16 \cos A \cos B \cos C+4 \cos ^2A+4 \cos ^2B+4 \cos ^2C-1\]
使用$\cos ^{2}A+\cos ^{2}B+\cos ^{2}C=-2\cos A \cos B \cos C +1$ "conditional" identities for the case α + β + γ = 180°
得$16 \cos A \cos B \cos C-8\cos A \cos B \cos C +3=0$
等价于$\cos A\cos B\cos C=-\frac38$

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