Similar Triangles and Centroids

Geometry Level 3

Triangle AOBAOB and triangle BOCBOC are directly similar (they have the same orientation). Let the centroids of these two triangles be M1M_1 and M2M_2, respectively.

If triangle M1OM2M_1OM_2 is right, and M1M2=ABM_1M_2=AB, then the sum of all possible values of cos2(ABO)\cos^2(\angle ABO) can be expressed as pq\dfrac{p}{q} for positive coprime integers p,qp,q.

Find p+qp+q.

Note: We do not know which angle of M1OM2 \triangle M_1 O M_2 is the right angle.

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