Really amazing Theorem.

• Jun 23rd 2010, 12:45 PM
Chris11
Really amazing Theorem.
Let XYZ be a triangle. Then the area of XYZ is given by $A=\frac{1}{2}B+I-1$, where B is the number of boundry lattice points; I is the number of interior lattice points.
• Jun 23rd 2010, 12:47 PM
Ackbeet
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, 12:48 PM
Chris11
Sorry, I meant that B is the number of boundry lattice points
• Jun 23rd 2010, 12:50 PM
Ackbeet
Ok. Are you asking for help proving this theorem, or are you just posting it for our edification?
• Jun 23rd 2010, 01:39 PM
Chris11
I just posted it because I thought people might find it interesting.
• Jun 23rd 2010, 02:15 PM
Ackbeet
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, 02:16 PM
awkward
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.