Extend OP each direction to obtain a diameter of the circle.
The distance from P along this diameter to the nearest intersection with the circle is 1 - x and the distance to the other end is 1 + x.
From geometry, the products of the parts of two intersecting chords are equal, so
(PC)(PB) = (1 - x)(1 + x)
Why? Equality occurs if and only if the vertex A coincides with the endpoint of the diameter determined by OP.
Sooooo. . .
Have you seen this before? See Maximum of f(x) = (1-x)(1+x)(1+x)