BC→=kAD→=ka⃗\overrightarrow{\mathrm{BC}} = k\overrightarrow{\mathrm{AD}} = k\vec{a}BC=kAD=ka AM→=12b⃗\overrightarrow{\mathrm{AM}} = \frac{1}{2}\vec{b}AM=21b AN→=a⃗+12DN→=a⃗+12(−a⃗+b⃗+ka⃗)=1+k2a⃗+12b⃗\overrightarrow{\mathrm{AN}} = \vec{a} + \frac{1}{2}\overrightarrow{\mathrm{DN}} = \vec{a} + \frac{1}{2}\left(-\vec{a} + \vec{b} + k\vec{a}\right) = \frac{1+k}{2}\vec{a} + \frac{1}{2}\vec{b}AN=a+21DN=a+21(−a+b+ka)=21+ka+21b MN→=−12AB→+AN→=−12b⃗+1+k2a⃗+12b⃗=1+k2a⃗\overrightarrow{\mathrm{MN}} = -\frac{1}{2}\overrightarrow{\mathrm{AB}} + \overrightarrow{\mathrm{AN}} = -\frac{1}{2}\vec{b} + \frac{1+k}{2}\vec{a} + \frac{1}{2}\vec{b} = \frac{1+k}{2}\vec{a}MN=−21AB+AN=−21b+21+ka+21b=21+ka これはa⃗\vec{a}aの整数倍なのでMN→∥AD→\overrightarrow{\mathrm{MN}} \parallel \overrightarrow{\mathrm{AD}}MN∥AD