Click here for a problem description
Furthermore we might use GSP to sketch the relationship between the product (PA)(PB)(PC) and the various parameters:
Click the GSP icon for a sketch that illustrates the graph and diagram above.
From playing with the sketch we make the following observations:
Based on these observations and the hint provided in the problem consider the following:
It follows that (BP)(PC) = (1 - x)(1 + x) and hence with equality when AP = 1 + x.
It follows that it remains to maximize (1 + x)(1 + x)(1 - x).
From simple calculus it follows that:
This knowledge provides the basis for a contruction of the family of triangles for whic (AP)(BP)(CP) is a maximum.
Click the GSP icon for a GSP script that draws the family of triangles given a circle.