第24題
如右圖，\(H\)為\(\triangle ABC\)之垂心，\(\overline{BC}=a,\overline{AC}=b,\overline{AB}=c\)，若面積比\(\triangle ABH:\triangle BCH:\triangle ACH=1:2:3\)，則邊長比\(a:b:c=\)　　　。
[解答]
\(\displaystyle \Delta BCH:\Delta ACH:\Delta ABH=\tan A:\tan B:\tan C=2:3:1 \)
設 \(\displaystyle \tan A=2k, \tan B=3k, \tan C=k \)
又 \(\displaystyle A+B+C=\pi \)
所以 \(\displaystyle \tan A+\tan B+\tan C=(\tan A)(\tan B)(\tan C) \)
\(\displaystyle 6k=6k^3 \)
\(\displaystyle k=1 \)
\(\displaystyle \tan A=2, \tan B=3, \tan C=1 \)
\(\displaystyle \sin A=\frac{2}{\sqrt5}, \sin B=\frac{3}{\sqrt{10}}, \sin C=\frac{1}{\sqrt2} \)
\(\displaystyle a:b:c=2\sqrt2:3:\sqrt5 \)

\( AH=2R\cos A, BH=2R\cos B, CH=2R\cos C \)
\( \angle{AHB}+C=\pi, \angle{BHC}+A=\pi, \angle{CHA}+B=\pi \)
\( (BCH) : (ACH) : (ABH)=2R^2\cos B \cos C \sin A:2R^2\cos C \cos A \sin B:2R^2\cos A \cos B \sin C=\tan A:\tan B: \tan C \)

20題
已知圓\(O\)的半徑為1，\(P\)為圓外一動點，\(\overline{PA},\overline{PB}\)為該圓的兩條切線，\(A\)、\(B\)為兩切點，那麼\(\overline{PA}\cdot \overline{PB}\)的最小值為　　　。

去年(100)陽明，半徑是2