<Tri-2024-2121>, hint

1>, Let A(0,0), B(p,q), C(x,y) p,q are integer

Consider Rotation matrix and rotate B(p,q) around A into C(x,y) by 60 degrees,

then x, y  are irrational.

2> Suppose A(0,0), B(-p,q), C(p,q), p, q are integer.

Then d(A,BC)/BC  is q/2p which is rational.