# Circles and Chords

Geometry Level 5

A tangent is drawn on a circle at the point of contact $$X$$. On this tangent, a variable point $$P$$ is chosen. A line is then drawn through $$P$$ such that it intersects the circle at two points $$A$$ and $$B$$ (not necessarily distinct) where $$B$$ is further away from $$P$$. Another circle centred at $$P$$ with radius $$PB$$ is drawn, and the line $$BX$$ (extended if necessary) intersects it at the points $$B$$ and $$C$$.

It is given that $$BX = 8$$ and that $$XC = 12$$. Calculate the mean value of $$(AB)(PB)$$ as $$P$$ moves on the tangent.

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