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(A_{1}A_{2})(BA_{2}+A_{2}C) +..+(C_{1}C_{2})(AC_{2}+C_{2}B)>3/4

Source: IMAC Arhimede 2008 p2

May 4, 2019

geometrygeometric inequalityinequalitiesangle bisectorcircumcircle

Problem Statement

In the

ABC

triangle, the bisector of

A

intersects the

[BC]

at the point

A_ {1}

, and the circle circumscribed to the triangle

ABC

at the point

A_ {2}

. Similarly are defined

B_ {1}

and

B_ {2}

, as well as

C_ {1}

and

C_ {2}

. Prove that

\frac {A_{1}A_{2}}{BA_{2} + A_{2}C} + \frac {B_{1}B_{2}}{CB_{2} + B_{2}A} + \frac {C_{1}C_{2}}{AC_{2} + C_{2}B} \geq \frac {3}{4}