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Three circles with a common point of intersection

Source: Nordic MO 2010 Q2

April 21, 2013
geometry unsolvedgeometry

Problem Statement

Three circles ΓA\Gamma_A, ΓB\Gamma_B and ΓC\Gamma_C share a common point of intersection OO. The other common point of ΓA\Gamma_A and ΓB\Gamma_B is CC, that of ΓA\Gamma_A and ΓC\Gamma_C is BB, and that of ΓC\Gamma_C and ΓB\Gamma_B is AA. The line AOAO intersects the circle ΓA\Gamma_A in the point X≠OX \ne O. Similarly, the line BOBO intersects the circle ΓB\Gamma_B in the point Y≠OY \ne O, and the line COCO intersects the circle ΓC\Gamma_C in the point Z≠OZ \ne O. Show that ∣AY∣⋅∣BZ∣⋅∣CX∣∣AZ∣⋅∣BX∣⋅∣CY∣=1.\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.
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