第14題
如下圖(圖形中各線段之比例僅供參考,實際之比例敘述如後),
設\(\displaystyle \frac{\overline{AD}}{\overline{DB}}=\frac{1}{3}\)且\(\displaystyle \frac{\overline{BE}}{\overline{EC}}=\frac{1}{2}\)且\(\displaystyle \frac{\overline{AF}}{2}=\frac{\overline{FG}}{1}=\frac{\overline{GC}}{2}\),若\(\vec{BH}=\alpha \vec{BA}+\beta \vec{BC}\),則實數對\((\alpha,\beta)=\) 。
[解答]
另解
假設BH與AC交於M
by孟氏定理
(AF/FM)*(MH/HB)*(BD/DA)=1 --------(1)
(CG/GM)*(MH/HB)*(BE/EC)=1 --------(2)
(1)/(2) => GM/FM = 1/6 => GM:FM = 1:6
BM = 3/7BA + 4/7BC (向量)
又by (1) MH:HB = 1:7 => MH:MB=1:6
BC=7/6BM=7/6(3/7BA + 4/7BC)=1/2BA + 2/3BC (向量)