ISL 1988 Problem 88

Seven circles are given. That is, there are six circles inside a fixed circle, each tangent to the fixed circle and tangent to the two other adjacent smaller circles. If the points of contact between the six circles and the larger circle are, in order, $A_1, A_2, A_3, A_4, A_5$ and $A_6$ prove that $$A_1 A_2 \cdot A_3 A_4 \cdot A_5 A_6 = A_2 A_3 \cdot A_4 A_5 \cdot A_6 A_1.$$