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chords in a circle, show area inequality

Source: S&M 2002 2nd Grade P2

May 14, 2021
geometrygeometric inequalityareainequalities

Problem Statement

Points A0,A1,…,A2kA_0,A_1,\ldots,A_{2k}, in this order, divide a circumference into 2k+12k+1 equal arcs. Point A0A_0 is connected by chords to all the other points. These 2k2k chords divide the interior of the circle into 2k+12k+1 parts. These parts are alternately painted red and blue so that there are k+1k+1 red and kk blue parts. Show that the blue area is larger than the red area.
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