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# Triangle Centers

What do the orthocenter, centroid, and circumcenter share in common? They lie on the Euler line!

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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