# Approximations to Equilateral Triangles by Heronian Triangles

## 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 $193-194-195$ 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

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $14$