## Almost Equilateral Triangles

#### Problem

We shall define an `almost equilateral triangles` to be a triangle for which two sides are equal and the third differs by no more than one unit. The smallest such triangle with integral length sides and area is 5-5-6.

Prove that infintitely many `almost equilateral triangles` with integral length sides and area exist.

#### Solution

By the definition an `almost equilateral triangles` measuring `a-a-b` is isosceles.

Using the Pythagorean Theorem: `a`^{2} = (`b`/2)^{2} + `h`^{2}, so 4`a`^{2} = `b`^{2} + 4`h`^{2}.

As `b` = `a` 1, `b`^{2} = `a`^{2} 2`a` + 1

4`a`^{2} = `a`^{2} 2`a` + 1 + 4`h`^{2}

3`a`^{2} 2`a` 1 + 4`h`^{2} = 0

9`a`^{2} 6`a` 3 12`h`^{2} = 0

9`a`^{2} 6`a` + 1 12`h`^{2} = 4

(3`a` 1)^{2} 12`h`^{2} = 4

((3`a` 1)/2)^{2} 3`h`^{2} = 1

By writing `x` = (3`a` 1)/2 and `y` = `h`, we get the Pell equation: `x`^{2} 3`y`^{2} = 1. Given one solution, it is well known that Pell equations have infinitely many solutions, and for completeness we shall prove this.

However, we must first show that integer `x` corresponds to an integer solutions for `a`; `b` being integer follows as `b` = `a` 1.

As `x` = (3`a` 1)/2, we get `a` = (2`x`1)/3

It should be clear that `x` cannot be divisible by 3, otherwise `x`^{2} 3`y`^{2} would be a multiple of 3 and could not be equal to 1.

So given that `x` 1 mod 3, 2`x` 1 mod 3, and so one of 2`x`+1 or 2`x`1 will be a multiple of 3. Hence for every integer solution of the equation `x`^{2} 3`y`^{2} = 1, we have an integer solution for `a` and `b`. Now we shall prove that infinitely many solutions exist.

Given (`x`,`y`), a solution pair to the Pell equation `x`^{2} 3`y`^{2} = 1, consider the larger pair (`x`^{2}+3`y`^{2},2`xy`):

(x^{2}+3y^{2})^{2} 3(2xy)^{2} | = | x^{4} + 6x^{2}y^{2} + 9y^{4} 12x^{2}y^{2} |

= | x^{4} 6x^{2}y^{2} + 9y^{4} | |

= | (x^{2} 3y^{2})^{2} | |

= | 1 |

In other words, if (`x`,`y`) is a solution then (`x`^{2}+3`y`^{2},2`xy`) will also be a solution, and as (7,4) leads to the first solution 5-5-6, we prove that infinitely many `almost equilateral triangles` with integral length sides and area exist.

Note that although the infinite solution set of the Pell equation `x`^{2} 3`y`^{2} = 1 is in a one-to-one mapping with the set of `almost equilateral triangles` we are seeking, this particular iterative method: (`x`,`y`) (`x`^{2}+3`y`^{2},2`xy`), will NOT produce every solution.