1986 CMO 第 4 题

Given a $\triangle ABC$ with its area equal to $1$ , suppose that the vertices of quadrilateral $P_1P_2P_3P_4$ all lie on the sides of $\triangle ABC$ . Show that among the four triangles $\triangle P_1P_2P_3, \triangle P_1P_2P_4, \triangle P_1P_3P_4, \triangle P_2P_3P_4$ there is at least one whose area is not larger than $1/4$ .

给定一个 $\triangle ABC$ ，其面积等于 $1$ ，假设四边形 $P_1P_2P_3P_4$ 的顶点都位于 $\triangle ABC$ 的边上。证明四个三角形 $\triangle P_1P_2P_3, \triangle P_1P_2P_4, \triangle P_1P_3P_4, \triangle P_2P_3P_4$ 中至少有一个其面积不大于 $1/4$ 。