Geometry
# Triangle Centers

In triangle \(ABC\), \(O\) is the circumcenter and \(I\) is the incenter. \(P\) is a point on the exterior angle bisector of \(\Delta BAC\) such that \( PI \perp OI \). Extend \(PI\) to intersect \(BC \) at \(Q\).

Find \(\dfrac{PI}{QI}\).

**Note**: Figure not drawn to scale.

If the vertices of a triangle have rational coordinates, then the coordinates of which of the following are **necessarily** rational?

**(A)** Centroid

**(B)** Circumcenter

**(C)** Orthocenter

**(D)** Incenter

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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