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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: a2 = (b/2)2 + h2, so 4a2 = b2 + 4h2.

As b = a plus or minus 1, b2 = a2 plus or minus 2a + 1

therefore 4a2 = a2 plus or minus 2a + 1 + 4h2
    3a2 plus or minus 2a minus 1 + 4h2 = 0
    9a2 plus or minus 6a minus 3 minus 12h2 = 0
    9a2 plus or minus 6a + 1 minus 12h2 = 4
    (3a plus or minus 1)2 minus 12h2 = 4
therefore ((3a plus or minus 1)/2)2 minus 3h2 = 1

By writing x = (3a plus or minus 1)/2 and y = h, we get the Pell equation: x2 minus 3y2 = 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 plus or minus 1.

As x = (3a plus or minus 1)/2, we get a = (2xplus or minus1)/3

It should be clear that x cannot be divisible by 3, otherwise x2 minus 3y2 would be a multiple of 3 and could not be equal to 1.

So given that x congruent plus or minus1 mod 3, 2x congruent plus or minus 1 mod 3, and so one of 2x+1 or 2xminus1 will be a multiple of 3. Hence for every integer solution of the equation x2 minus 3y2 = 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 x2 minus 3y2 = 1, consider the larger pair (x2+3y2,2xy):

(x2+3y2)2 minus 3(2xy)2 = x4 + 6x2y2 + 9y4 minus 12x2y2
  = x4 minus 6x2y2 + 9y4
  = (x2 minus 3y2)2
  = 1

In other words, if (x,y) is a solution then (x2+3y2,2xy) 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 x2 minus 3y2 = 1 is in a one-to-one mapping with the set of almost equilateral triangles we are seeking, this particular iterative method: (x,y) maps (x2+3y2,2xy), will NOT produce every solution.

Problem ID: 219 (30 Mar 2005)     Difficulty: 4 Star

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