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Special Points on $BC$

Source: 2013 USAMO Problem 6

May 1, 2013
geometrycircumcirclegeometric transformationreflectionratiotrigonometryAsymptote

Problem Statement

Let ABCABC be a triangle. Find all points PP on segment BCBC satisfying the following property: If XX and YY are the intersections of line PAPA with the common external tangent lines of the circumcircles of triangles PABPAB and PACPAC, then (PAXY)2+PB⋅PCAB⋅AC=1.\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.
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