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# Triangle Centers

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

Triangle \(ABC\) has centroid \(G\). Triangles \(ABG\), \(BCG\), and \(CAG\) have centroids \(G_1,G_2,G_3\) respectively.

The value of \(\dfrac{[G_1G_2G_3]}{[ABC]}\) can be represented by \(\dfrac{p}{q}\), for positive coprime integers \(p,q\).

Find \(p+q\).

\(\text{Details and Assumptions:}\)

\([ABCD]\) denotes the area of \(ABCD\).

How many triangles \(ABC\) are there with integer side lengths such that the area of the triangle formed by joining the orthocenter, the circumcenter and the centroid of \(\triangle ABC\) is \(44\) square units?

**Details and assumptions:**

The orthocenter of \(ABC\) is the point at which the altitudes of \(ABC\) intersect.

The circumcenter of \(ABC\) is the point which is equidistant from \(A\), \(B\) and \(C\).

The centroid of \(ABC\) is the point at which the medians of \(ABC\) intersect.

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