Geometry
# Triangle Centers

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 \(p+q\).

Let \(ABC\) be a triangle in which \(AB=AC\). Suppose the orthocentre of the triangle lies on the incircle.

Find \(\dfrac{AB}{BC}\).

Consider \( \Delta ABC \) such that \(AB = 13, BC = 14, AC = 15 \)

Let \(AD \perp BC, BE \perp AC, CF \perp AB \) and let \(H\) be its orthocenter

Let \(R(ABC)\) denote circumradius of \( \Delta ABC \)

Let \(r_0 \) be inradius and \(r_1, r_2, r_3 \) be the exradii of \( \Delta ABC \)

Then find the value of

\[ R(ABC) + R(HAB) + R(HBC) + R(HAC) + \sum_{i=0}^3 r_i \]

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