Areas in Triangle II

Here's a problem very much related to a better known one. Given a triangle, extend each side by its own length in one direction following a certain order. Connect the newly obtained points. Prove that the area of the big triangle is 7 times that of the given one.

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Of course, we can add 1, 2, 3, or, in general, N lengths to each side. The triangles so obtained have areas 7, 19, 37, or, in general, 3N² + 3N + 1 times greater than the given triangle. (This this #A003215 in the On-Line Encyclopedia of Integer Sequences.)

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