Let XYZ be a triangle. Then the area of XYZ is given by , where B is the number of boundry lattice points; I is the number of interior lattice points.

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- Jun 23rd 2010, 01:45 PMChris11Really amazing Theorem.
Let XYZ be a triangle. Then the area of XYZ is given by , where B is the number of boundry lattice points; I is the number of interior lattice points.

- Jun 23rd 2010, 01:47 PMAckbeet
Both B and I are the number of interior lattice points? I know virtually nothing about this theorem, but that does strike me as a bit odd.

- Jun 23rd 2010, 01:48 PMChris11
Sorry, I meant that B is the number of boundry lattice points

- Jun 23rd 2010, 01:50 PMAckbeet
Ok. Are you asking for help proving this theorem, or are you just posting it for our edification?

- Jun 23rd 2010, 02:39 PMChris11
I just posted it because I thought people might find it interesting.

- Jun 23rd 2010, 03:15 PMAckbeet
That's cool. I think it would be even more interesting if you were to outline the construction of the lattice. I've seen a result or two similar to this, I think, in number theory. If it's what I'm thinking of.

- Jun 23rd 2010, 03:16 PMawkward
This is the special case (for triangles) of Pick's Theorem. See

Pick's theorem - Wikipedia, the free encyclopedia

However, you left out a condition: The vertices of the triangle (or polygon, in the general case) must lie on grid points.

I agree with you, it's a fascinating theorem.