Solve a problem involving complex numbers

[Gavran]

It is all about a $30^\circ-60^\circ-90^\circ$ triangle whose three angles are in the ratio $1:2:3$ and whose three sides are in the ratio $1:\sqrt 3:2$. This triangle and its reflection over the hypotenuse form a right kite with two other angles of $60^\circ$ and $120^\circ$.