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Equality with circumradius [ILL 1974]

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January 2, 2011
geometrycircumcircleEulertrigonometrygeometry unsolved

Problem Statement

Through the circumcenter OO of an arbitrary acute-angled triangle, chords A1A2,B1B2,C1C2A_1A_2,B_1B_2, C_1C_2 are drawn parallel to the sides BC,CA,ABBC,CA,AB of the triangle respectively. If RR is the radius of the circumcircle, prove that A1O⋅OA2+B1O⋅OB2+C1O⋅OC2=R2.A_1O \cdot OA_2 + B_1O \cdot OB_2 + C_1O \cdot OC_2 = R^2.
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