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An equilateral triangle

Source: Iberoamerican Olympiad 1992, Problem 3

May 14, 2007
geometrycircumcircletrigonometryinradiusinequalitiesincentergeometry proposed

Problem Statement

Let ABCABC be an equilateral triangle of sidelength 2 and let ω\omega be its incircle. a) Show that for every point PP on ω\omega the sum of the squares of its distances to AA, BB, CC is 5. b) Show that for every point PP on ω\omega it is possible to construct a triangle of sidelengths APAP, BPBP, CPCP. Also, the area of such triangle is 34\frac{\sqrt{3}}{4}.
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