Kinematic proof Fermat point
Observe that the velocities of points F and D have the same magnitude, but are rotated 120˚ with respect to each other. Thus, vector AD is equal to vector BD rotated 120˚ plus a constant vector. To see that the constant vector is zero, we can start with ABC as an equilateral triangle.
Drag C quickly and observe the separation of points on the traces of F, C, and D.