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\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3} area cevian triangle inequality

Source: Sharygin 2005 X,XI CR 14

August 19, 2019
geometryarea of a triangleareasgeometric inequalitytriangle inequalityinequalities

Problem Statement

Let PP be an arbitrary point inside the triangle ABCABC. Let A1,B1A_1, B_1 and C1C_1 denote the intersection points of the straight lines AP,BPAP, BP and CPCP, respectively, with the sides BC,CABC, CA and ABAB. We order the areas of the triangles AB1C1,A1BC1,A1B1CAB_1C_1,A_1BC_1,A_1B_1C. Denote the smaller by S1S_1, the middle by S2S_2, and the larger by S3S_3. Prove that S1S2≤S≤S2S3\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3} ,where SS is the area of the triangle A1B1S1A_1B_1S_1.
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