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Find the greatest possible value of $\frac{\min\{S(BPB_1),S(CPC_1)\}}{S(ABC)}$

Source: Moldova TST 2000

August 7, 2023

geometry

Problem Statement

In triangle

ABC

points

B_1

and

C_1

are on

AB

and

AC

respectively and

P{}

is a point on the segment

B_1C_1

. Find the greatest possible value of

\frac{\min\{S(BPB_1),S(CPC_1)\}}{S(ABC)}

, where

S(XYZ)

is the area o the triangle

ABC