Approximations to Equilateral Triangles by Heronian Triangles

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Theorem

The sequence of best approximations to an equilateral triangle by a Heronian triangle begins:

The $\left({3, 4, 5}\right)$ triangle, with area $6$
The $13-14-15$ triangle, with area $84$, where $14 = 4^2 - 2$
The $193-194-195$ triangle, where $194 = 14^2 - 2$
The $37 \, 633-37 \, 634-37 \, 635$ triangle, where $37 \, 634 = 194^2 - 2$

and so on.

This sequence is A003010 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


Sources