MathDB

Let

ABC

be an acute triangle with

\omega,S

, and

R

being its incircle, circumcircle, and circumradius, respectively. Circle

\omega_{A}

is tangent internally to

S

A

and tangent externally to

\omega

. Circle

S_{A}

is tangent internally to

S

A

and tangent internally to

\omega

. Let

P_{A}

and

Q_{A}

denote the centers of

\omega_{A}

and

S_{A}

, respectively. Define points

P_{B}, Q_{B}, P_{C}, Q_{C}

analogously. Prove that

8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; ,

with equality if and only if triangle

ABC

is equilateral.

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