How Well Do You Know Your Geometry?

Geometry Level 2

How many triangles ABCABC are there with integer side lengths such that the area of the triangle formed by joining the orthocenter, the circumcenter and the centroid of ABC\triangle ABC is 4444 square units?


Details and assumptions:

The orthocenter of ABCABC is the point at which the altitudes of ABCABC intersect.

The circumcenter of ABCABC is the point which is equidistant from AA, BB and CC.

The centroid of ABCABC is the point at which the medians of ABCABC intersect.

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