1961 IMO 第 4 题

In the interior of $\triangle P_{1} P_{2} P_{3}$ a point $P$ is given. Let $Q_{1}, Q_{2}$, and $Q_{3}$ respectively be the intersections of $P P_{1}, P P_{2}$, and $P P_{3}$ with the opposing edges of $\triangle P_{1} P_{2} P_{3}$. Prove that among the ratios $P P_{1} / P Q_{1}, P P_{2} / P Q_{2}$, and $P P_{3} / P Q_{3}$ there exists at least one not larger than 2 and at least one not smaller than 2.

在$\三角形 P_{1} P_{2} P_{3}$ 的内部给出一个点$P$。令$Q_{1}、Q_{2}$和$Q_{3}$分别为$P P_{1}、P P_{2}$和$P P_{3}$与$\三角形P_{1} P_{2} P_{3}$的相对边的交点。证明比率 $P P_{1} / P Q_{1}、P P_{2} / P Q_{2}$ 和 $P P_{3} / P Q_{3}$ 中存在至少一个不大于 2 且至少存在一个不小于 2。