4. A triangle $A B C$ with angle $A$ equal to $60^{\circ}$ is given. Its incircle is tangent to side $A B$ at point $D$, while its excircle tangent to side $A C$, is tangent to the extension of side $A B$ at point $E$. Prove that the perpendicular to side $A C$, passing through point $D$, meets the incircle again at a point equidistant from points $E$ and $C$. (The excircle is the circle tangent to one side of the triangle and to the extensions of two other sides.) Azamat Mardanov