내분·외분과 넓이 비

[출제의도] 선분의 내분점과 외분점의 위치를 찾을 있는가를 묻는 문제이다.

$\displaystyle \overline{{\mathrm{A}} \mathit{\mathrm{P}}} : \overline{{\mathrm{P}} \mathit{\mathrm{Q}}} = 2 : 1$이므로 $\displaystyle \triangle {\mathrm{C}} \mathit{\mathrm{P}} \mathit{\mathrm{Q}} \mathit{=} \frac{1}{2} \triangle {\mathrm{A}} \mathit{\mathrm{P}} \mathit{\mathrm{C}}$, $\displaystyle \triangle {\mathrm{A}} \mathit{\mathrm{P}} \mathit{\mathrm{C}} \mathit{=} \frac{1}{4} \triangle {\mathrm{A}} \mathit{\mathrm{B}} \mathit{\mathrm{C}}$에서 $\displaystyle \triangle {\mathrm{C}} \mathit{\mathrm{P}} \mathit{\mathrm{Q}} \mathit{=} \frac{1}{8} \triangle {\mathrm{A}} \mathit{\mathrm{B}} \mathit{\mathrm{C}}$ $\therefore$ $\displaystyle \frac{\triangle {\mathrm{A}} \mathit{\mathrm{B}} \mathit{\mathrm{C}}}{\triangle {\mathrm{C}} \mathit{\mathrm{P}} \mathit{\mathrm{Q}}} = 8$