Mock AIME 2 2006-2007 Problems/Problem 9

Problem

In right triangle $\displaystyle ABC,$ $\displaystyle \angle C=90^\circ.$ Cevians $\displaystyle AX$ and $\displaystyle BY$ intersect at $\displaystyle P$ and are drawn to $\displaystyle BC$ and $\displaystyle AC$ respectively such that $\displaystyle \frac{BX}{CX}=\frac23$ and $\displaystyle \frac{AY}{CY}=\sqrt 3.$ If $\displaystyle \tan \angle APB= \frac{a+b\sqrt{c}}{d},$ where $\displaystyle a,b,$ and $\displaystyle d$ are relatively prime and $\displaystyle c$ has no perfect square divisors excluding $\displaystyle 1,$ find $\displaystyle a+b+c+d.$

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Mock AIME 2 2006-2007 Problems/Problem 9

Problem