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Omni-MATH / 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 …

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problem

For a triangle ABC, ABC, let k k be its circumcircle with radius r. r. The bisectors of the inner angles A,B, A, B, and C C of the triangle intersect respectively the circle k k again at points $ A', B',$ and $ C'.$ Prove the inequality 16Q3≥27r4P, 16Q^3 \geq 27 r^4 P, where Q Q and P P are the areas of the triangles $ A'B'C'$ and ABCABC respectively.
Plain-text mathematical notation (without MathML)
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

16Q³≥27r⁴P,

where Q and P are the areas of the triangles $ A'B'C'$ and ABC respectively.
Original LaTeX notation
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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task_id
babf1d5e-11e3-5150-a263-724133362d54
task_key
test--babf1d5e-11e3-5150-a263-724133362d54
task_revision_id
1
config / split
/ test

Official source

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