Prove that, if we choose nine vertices of a regular icosagon, some three of these will form an isosceles triangle.
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Label the vertices, in order, A, B, C, D, A, B, C, D, etc. Now each label defines a regular pentagon. Of our nine chosen points, at least three must bear the same label (two of each label would account for only eight points). So we have three vertices of a regular pentagon. Since there are only two possible distances between a pentagon’s vertices, the distances between two pairs of these points must be equal.
By Polish mathematician Paul Vaderlind.
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