Radiohead (2+2=5)

Geometry Level 4

The altitudes \(BE\) and \(CF\) of a triangle \(ABC\) meet at \(H\). The extensions of \(FE\) and \(BC\) meet at \(U\). A line through the midpoint \(L\) of \(BC\) parallel to the internal bisector of \(\angle{EUB}\) meets \(CA, AB, HC, HB\) (extended as necessary) at \(P, Q, X, Y\), respectively. If the circumradius of \(APQ\) is \(r_{1}\) and the circumradius of \(HXY\) is \(r_{2}\), find \(\dfrac{r_{1}}{r_{2}}\).

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