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The ratio AB·BC·CA/R inequality [ILL 1971]

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January 1, 2011
ratioinequalitiesanalytic geometrygeometrycircumcirclegeometry proposed

Problem Statement

Let A,B,CA,B,C be three points with integer coordinates in the plane and KK a circle with radius RR passing through A,B,CA,B,C. Show that AB⋅BC⋅CA≥2RAB\cdot BC\cdot CA\ge 2R, and if the centre of KK is in the origin of the coordinates, show that AB⋅BC⋅CA≥4RAB\cdot BC\cdot CA\ge 4R.
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