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第 11 題:

  

如圖,在 \(\overline{AP},\overline{DP}\) 上分別取點 \(E,F\) 使得 \(\overline{EB}\perp\overline{PB}, \overline{FC}\perp\overline{PC}\)

則 \(\displaystyle\tan\alpha\times\tan\beta=\frac{\overline{EB}}{\overline{PB}}\times\frac{\overline{CF}}{\overline{PC}}\)

       \(\displaystyle=\frac{\overline{EB}}{\overline{PC}}\times\frac{\overline{CF}}{\overline{PB}}\)(注意:\(\overline{EB}\Big/\Big/\overline{PC}\) 且 \(\overline{CF}\Big/\Big/\overline{PB}\))

       \(\displaystyle=\frac{\overline{AB}}{\overline{AC}}\times\frac{\overline{DC}}{\overline{DB}}\)

       \(\displaystyle=\frac{3}{3+5}\times\frac{4}{5+4}\)

       \(\displaystyle=\frac{1}{6}.\)

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