Centroid, Incentre, Orthocentre?

Geometry Level 5

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

cos2ABC=pq+r,\cos^2 \angle ABC = \dfrac{p}{q+\sqrt{r}}, where pp and qq are positive integers and rr is a prime number. Find the value of p+q+rp+q+r.

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