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 $13-14-15$ 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.