# Similar Triangles and Centroids

Geometry Level 3

Triangle $AOB$ and triangle $BOC$ are directly similar (they have the same orientation). Let the centroids of these two triangles be $M_1$ and $M_2$, respectively.

If triangle $M_1OM_2$ is right, and $M_1M_2=AB$, then the sum of all possible values of $\cos^2(\angle ABO)$ can be expressed as $\dfrac{p}{q}$ for positive coprime integers $p,q$.

Find $p+q$.

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