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Show that sum of areas is equal

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September 14, 2010
geometrytrigonometryratiosimilar trianglesgeometric sequencegeometry proposed

Problem Statement

Let Γi,i=0,1,2,…\Gamma_i, i = 0, 1, 2, \dots , be a circle of radius rir_i inscribed in an angle of measure 2α2\alpha such that each Γi\Gamma_i is externally tangent to Γi+1\Gamma_{i+1} and ri+1<rir_{i+1} < r_i. Show that the sum of the areas of the circles Γi\Gamma_i is equal to the area of a circle of radius r=12r0(sin⁡α+cscα).r =\frac 12 r_0 (\sqrt{ \sin \alpha} + \sqrt{\text{csc} \alpha}).
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