I am hoping a few of you mathematicians more experienced with writing proofs might give me some guidance here and possibly give me some ideas about how to restructure the following into a more rigorous statement.
- Prove that there does not exist an equilateral triangle in the plane whose vertices are at integer lattice points (x,y).
My response follows:
Working with an equilateral triangle symmetrical about the y-axix, Let the height of the triangle along the y-axis be B, let the base of the triangle be A and let the two remaining sides both be C.
Notice this forms two triangles where, in each triangle, C=2A. This allows us to rearrange Pythagorus' theorem such that b=Sqrt(c^2 - (2c)^2) which, after a bit of hand waving, becomes B=(sqrt(3)*sqrt(c^2))/2.
Since B cannot equal any integer value for any value of C, B cannot reside on a integer lattice point in the plane. Thus, there exists no equilateral triangle in the plane with vertices at integer lattice points (x,y).
This is day two of class. We've yet to have any formal training in proof writing. I really do welcome and appreciate any and all advice!
Hint: Assume there is a equilateral triangle whose vertices are all lattice points. Then, look at the area of the triangle using the formula $A = \dfrac{s^2\sqrt{3}}{4}$, where $s$ is the side length. Also, look at the area of the triangle using Pick's Theorem. Do you see a contradiction?