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6.角GCP=角GBP=角GBA

PB+PC=2Y+(X-Y)=X+Y=PA

"一個圓的兩條弦"那題，題目有問題，BE不唯一

2
\begin{align} & {{T}_{n}}=\frac{2}{3}\times \frac{4}{3}\times \frac{4}{5}\times \frac{6}{5}\times \cdots \cdots \times \frac{2n}{2n+1}\times \frac{2n+2}{2n+1} \\ & {{T}_{n}}^{2}>\frac{1}{2}\times \frac{3}{2}\times \frac{2}{3}\times \frac{4}{3}\times \frac{3}{4}\times \frac{5}{4}\times \cdots \cdots \times \frac{2n-1}{2n}\times \frac{2n+1}{2n}\times \frac{2n}{2n+1}\times \frac{2n+2}{2n+1}=\frac{n+1}{2n+1} \\ & {{T}_{n}}^{2}<\frac{2}{3}\times \frac{4}{3}\times \frac{3}{4}\times \frac{5}{4}\times \frac{4}{5}\times \frac{6}{5}\times \cdots \cdots \times \frac{2n}{2n+1}\times \frac{2n+2}{2n+1}\times \frac{2n+1}{2n+2}\times \frac{2n+3}{2n+2}=\frac{2n+3}{3n+3} \\ \end{align}

2
\begin{align} & {{x}^{2}}+2xy-1=0 \\ & 2{{y}^{2}}+xy-2=0 \\ & 2\left( {{x}^{2}}+2xy \right)=2{{y}^{2}}+xy=2 \\ & 2{{x}^{2}}+3xy-2{{y}^{2}}=0 \\ & \left( x+2y \right)\left( 2x-y \right)=0 \\ & y=2x \\ & ...... \\ & \\ \end{align}

2
\begin{align} & {{x}^{2}}+xy+{{y}^{2}}-1=0 \\ & {{y}^{2}}-4\left( {{y}^{2}}-1 \right)\ge 0 \\ & ...... \\ \end{align}

15
\begin{align} & x\left( {{x}^{2}}-yz \right)=3 \\ & y\left( {{y}^{2}}-xz \right)=9 \\ & z\left( {{z}^{2}}-xy \right)=30 \\ & \\ & xyz={{x}^{3}}-3={{y}^{3}}-9={{z}^{3}}-30 \\ & y=\sqrt[3]{{{x}^{3}}+6},z=\sqrt[3]{{{x}^{3}}+27} \\ & \\ & x\sqrt[3]{{{x}^{3}}+6}\sqrt[3]{{{x}^{3}}+27}={{x}^{3}}-3 \\ & {{x}^{3}}\left( {{x}^{3}}+6 \right)\left( {{x}^{3}}+27 \right)={{\left( {{x}^{3}}-3 \right)}^{3}} \\ & 42{{x}^{6}}+135{{x}^{3}}+27=0 \\ & \left( {{x}^{3}}+3 \right)\left( 42{{x}^{3}}+9 \right)=0 \\ & {{x}^{3}}=-3\ or\ -\frac{3}{14} \\ & xyz=-6\ or\ -\frac{45}{14} \\ & \left( x,y,z \right)=\left( -\sqrt[3]{3},\sqrt[3]{3},2\sqrt[3]{3} \right) \\ \end{align}

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https://math.pro/db/attachment.php?aid=2693&k=d33bc06937b48c04308f4f8d0a35bde0&t=1627085999

∠BAC 用銳角和鈍角分別畫圖，就知道 BE 不唯一了

https://math.pro/db/attachment.php?aid=3807&k=7690112094de3c74cc18b662ceb6808d&t=1627085999

6.

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