Editing Equilateral triangle (section)

=== Relationship with circles ===
The radius of the [[circumscribed circle]] is:
<math display="block"> R = \frac{a}{\sqrt{3}}, </math>
and the radius of the [[incircle and excircles of a triangle|inscribed circle]] is half of the circumradius:
<math display="block"> r = \frac{\sqrt{3}}{6}a. </math>

A [[Euler's theorem in geometry|theorem of Euler]] states that the distance <math> t </math> between circumcenter and incenter is formulated as <math> t^2 = R(R - 2r) </math>. As a corollary of this, the equilateral triangle has the smallest ratio of the circumradius <math>R</math> to the inradius <math>r</math> of any triangle. That is:{{sfnp|Svrtan|Veljan|2012}}
<math display="block"> R \ge 2r. </math>

[[Pompeiu's theorem]] states that, if <math>P</math> is an arbitrary point in the plane of an equilateral triangle <math>ABC</math> but not on its [[circumcircle]], then there exists a triangle with sides of lengths <math>PA</math>, <math>PB</math>, and <math>PC</math>. That is, <math>PA</math>, <math>PB</math>, and <math>PC</math> satisfy the [[triangle inequality]] that the sum of any two of them is greater than the third. If <math>P</math> is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as [[Van Schooten's theorem]].{{sfnp|Alsina|Nelsen|2010|p=[https://books.google.com/books?id=mIT5-BN_L0oC&pg=PA102 102&ndash;103]}}

A [[packing problem]] asks the objective of [[Circle packing in an equilateral triangle|<math> n </math> circles packing into the smallest possible equilateral triangle]]. The optimal solutions show <math> n < 13 </math> that can be packed into the equilateral triangle, but the open conjectures expand to <math> n < 28 </math>.{{sfnp|Melissen|Schuur|1995}}