SHARYGIN 2005 Problem 14

Let $P$ be an arbitrary point inside the triangle $ABC$ . Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$ , respectively, with the sides $BC, CA$ and $AB$ . We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$ . Denote the smaller by $S_1$ , the middle by $S_2$ , and the larger by $S_3$ . Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$ .