For concyclic $A$, $B$, $C$, $D$ with $AB=BC=CD$, if bisectors of $\angle ABD$ and $\angle ACD$ meet at $E$, and $AE\parallel CD$, find $\angle ABC$ An Olympiad geometry problem:

Let $A, B, C$ and $D$ be points on a circle such that $AB = BC = CD$. The angle bisectors of $\angle ABD$ and $\angle ACD$ intersect at the point $E$. If the lines $AE$ and $CD$
are parallel, find $\angle ABC$.

I worked on this question and found that $E$ also should be situated in the same circle. And then I was able to prove that $AECD$ was a parallelogram using congruent triangles $ACE$ and $DCE$.I took Angle $DCE$ as $y$ and angle $CDB$ as $x$ then the required angle is $2y+x$. Finally, I got $y=45$ and $x=30$.Therefore the required angle is $120$. But these values make $AD$ The diameter of the circle.Then there is a way more easier proof for this problem by reflecting it along $AD$ and we get a regular hexagon which makes $ABC$ $120$.
Is there a way to prove $AD$ is the diameter of circle straight away or is my proof wrong?
 A: 
There are, up to symmetries, two possible configurations. As OP suggested, these correspond to a regular convex heptagon or a regular star heptagon with 2 missing vertices.
Since $AB=BC=CD$, the corresponding arcs $\overset{\huge\frown}{AB},\overset{\huge\frown}{BC}$ and $\overset{\huge\frown}{CD}$ are equal. Let $x$ denote their measure. Let $E_1$ be the midpoint of arc $\overset{\huge\frown}{AD}$ (not containing points $B$ and $C$). Since $\overset{\huge\frown}{AE_1}=\overset{\huge\frown}{E_1D}$, $\angle ACE_1=\angle DCE_1$ so $CE_1$ is the angle bisector of $\angle ACD$. Similarly, $\angle ABE_1=\angle DBE_1$ so $BE_1$ is the angle bisector of $\angle ABD$. Thus, the two angle bisectors intersect at $E_1$ which means that $E_1$ is the same point as $E$. Since $AE\parallel CD$, we have $\angle AEC=\angle DCE$ which implies that $\overset{\huge\frown}{AC}=\overset{\huge\frown}{ED}$. Thus, $\overset{\huge\frown}{AE}=\overset{\huge\frown}{ED}=2x$ or $360°-2x$, and at this point every arc on the circle is expressed in terms of $x$. Can you see how to finish now?
