IMO 1974 LL USA41

Through the circumcenter O of an arbitrary acute-angled trian-

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IMO 1974 LL USA41

Origin: USA

Problem

Through the circumcenter O of an arbitrary acute-angled trian- gle, chords A1A2, B1B2, C1C2 are drawn parallel to the sides BC, CA, AB of the triangle respectively. If R is the radius of the circumcircle, prove that A1O \cdot OA2 + B1O \cdot OB2 + C1O \cdot OC2 = R2.