Distance between Two Fermat Points

Let and be the Fermat points [1] of the triangle ; is the point with minimum sum of distances to the vertices. Let be the circumradius of and be the Brocard angle [2] of .

The triangles and are the third vertices of equilateral triangles drawn on the outside and inside of .

Define:

(shown in green),

,

.

Then

.

You can drag the vertices , , .