One blog I follow regularly is Antonio Gutierrez's gogeometry. If you teach/study/like plane geometry he should be one of your regular references.
Recently among his posts have been a couple with a related theme, circles inscribed or circumscribed about an equilateral triangle. I'm listing these because they are each a wonderful relationship, and together give these otherwise somewhat mundane seeming triangles a luster students?teachers/others might miss.
I will post the problems, but not the proofs, which (if you can't/won't work them out yourself you can find at the links provided to Antonio's site.
So on we go...
1) draw a circle and inscribe an equilateral triangle. Now pick any point on the circumference and construct segments from this point to the three vertices. The sum of the lengths of the two shorter segments will equal the third. The problem, and solution is here.
Same triangle, same circle, but now we sum the square of the three distances ...????? and they sum to twice the square of a side of the equilateral triangle. That proof is here.
3) And now one with the circle on the inside. Again, from any point on the circle construct segments to the three vertices of the equilateral triangle. Again the sum of the squares is related to a side length, but I'll let you chase that down for yourself. Or you can go to the site here.
Addendum: John Golden sent a comment with a link to a GeoGebra sketch showing all three.