填充第 10 題
\(\displaystyle \left(x\sin\frac{\pi}{7}\right)^7=128\Rightarrow \left|x\right|=\frac{2}{\sin\frac{\pi}{7}}\)
\(\displaystyle \omega_0,\omega_1,\omega_2,\cdots,\omega_6\) 等同於位在以原點為圓心，
以 \(\displaystyle \frac{2}{\sin\frac{\pi}{7}}\) 為半徑的圓周上的七等分點，
令此七點依序為 \(A,B,C,D,E,F,G\)
由托勒蜜定理，可知 \(\displaystyle \overline{AC}\cdot\overline{BD}=\overline{AD}\cdot \overline{BC}+\overline{AB}\cdot \overline{CD}\)
\(\displaystyle \Rightarrow \overline{AC}^2=\overline{AD}\cdot \overline{AB}+\overline{AB}^2\)
\(\displaystyle \Rightarrow \overline{AC}^2-\overline{AD}\cdot \overline{AB}=\overline{AB}^2\)
而所求 \(\displaystyle =2 \overline{AC}^2-2\overline{AD}\cdot \overline{AB}=2\overline{AB}^2=2\cdot 4^2=32\)
註：
