IMO 1988 LL USS88

There are six circles inside a fixed circle, each tangent to

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IMO 1988 LL USS88

Origin: USS

Problem

There are six circles inside a fixed circle, each tangent to the fixed circle and tangent to the two adjacent smaller circles. If the points of contact between the six circles and the larger circle are, in order, A1, A2, A3, A4, A5, and A6, prove that A1A2 \cdot A3A4 \cdot A5A6 = A2A3 \cdot A4A5 \cdot A6A1.