MathDB

Bosnia and Herzegovina TST 2000 Day 1 Problem 2

Source: Bosnia and Herzegovina Team Selection Test 2000

September 19, 2018

geometrycircumcircleiff

Problem Statement

Let

S

be a point inside triangle

ABC

and let lines

AS

BS

and

CS

intersect sides

BC

CA

and

AB

in points

X

Y

and

Z

, respectively. Prove that

\frac{BX\cdot CX}{AX^2}+\frac{CY\cdot AY}{BY^2}+\frac{AZ\cdot BZ}{CZ^2}=\frac{R}{r}-1

iff

S

is incenter of

ABC