Inequality in a triangle

Let A, B, and C be the vertices of any triangle inscribed in a unit circle. Let P be a point on segment BC. As P moves along segment BC, what is the maximum value for


where PA, PB, and PC are the measures of the lengths of these three segments?

Note, this is a question for all possible triangles ABC inscribed in the circle. That is, of ALL inscribed triangles in a unit circle, what is the maximum this product, (PA)(PB)(PC) could obtain. It will not be the case that every inscribed triangle will have the maximum product; but of the set of all possible triangles, what is the maximum this product could reach?

Some possibly helpful ideas.    (Okay, there is no guarantee they are helpful . . .)

Construct a triangle inscribed in a unit circle such the maximum for (PA)(PB)(PC) is obtained.

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