Centroid, Incentre, Orthocentre?

Geometry Level 5

As shown above, an angle bisector, median, and altitude intersect at the same point in right triangle \(ABC.\)

\(\cos^2 \angle ABC = \dfrac{p}{q+\sqrt{r}},\) where \(p\) and \(q\) are positive integers and \(r\) is a prime number. Find the value of \(p+q+r\).

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