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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