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16Q^3 >= 27r^4P

Source: IMO Shortlist 1989, Problem 6, ILL 14

September 18, 2008

geometrycircumcirclegeometric inequalityarea of a triangleIMO Shortlist

Problem Statement

For a triangle

ABC,

let

k

be its circumcircle with radius

r.

The bisectors of the inner angles

A, B,

and

C

of the triangle intersect respectively the circle

k

again at points

A', B',

and

C'.

Prove the inequality

16Q^3 \geq 27 r^4 P,

where

Q

and

P

are the areas of the triangles

A'B'C'

and

ABC

respectively.

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