Take a point $P$ on a side of $ABC$ and rotate $ABC$ around $P$ by $60^\circ$ clockwise/counterclockwise. The intersections between the sides of the rotated triangle and the original triangle provide two points $Q,R$ such that $PQR$ is equilateral. In follows that there are infinite equilateral triangles inscribed in a given triangle. On the other hand, if the sides length is fixed, there are at most two solutions. I am going to provide a proof almost-without words:
The red locus is convex (since it is the arc of an ellipse centered at the upper vertex), hence it cannot meet the horizontal side at more than two points.