Circle and Equilateral
Pat Bellew pointed out three related results from gogeometry.com
in this post: http://pballew.blogspot.com/2014/01/circles-and-equilateral-triangles.html
1: http://www.gogeometry.com/problem/p256_equilateral_triangle_circumcircle_distance_vertex.htm
2: http://www.gogeometry.com/problem/p257_equilateral_triangle_circumcircle_distance_sides.htm
3: http://www.gogeometry.com/problem/p258_equilateral_triangle_incircle_distance_square.htm