Plane rotation lesson

Question

Given a point $A$ and two parallel lines $(D)$ and $(D')$ not passing through $A$. Construct a point $B$ on $(D)$ and a point $C$ on $(D')$ such that $\Delta ABC$ is direct equilateral.

Answer 1

Let $R_A^{60^{\circ}}$ be a rotation by $60^{\circ}$ around the point $A$.

Let $R_A^{60^{\circ}}((D'))\cap (D)=\{B\}$ and $R_A^{-60^{\circ}}(B)=\{C\}$.

Thus, $\Delta ABC$ is an equilateral triangle.