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 \(\overline{CF}\Big/\Big/\overline{PB}\))
\(\displaystyle=\frac{\overline{AB}}{\overline{AC}}\times\frac{\overline{DC}}{\overline{DB}}\)
\(\displaystyle=\frac{3}{3+5}\times\frac{4}{5+4}\)
\(\displaystyle=\frac{1}{6}.\)