Eight Solutions

Here are the starting points and sketches for eight solutions.

1. Draw segment $DF$ parallel to $BC$ with $F$ on $AB$. Draw $CF$ intersecting $BD$ at $G$. Now find the equilateral triangles and isosceles triangles.
2. Use the Law of Sines in triangle $BED$ and triangle $BCD$. Use $BE=BC$ to connect the results. Simplify and solve for $\angle EDB$.
3. Draw lines through $D$ and $B$ parallel to $BC$ and $DC$, respectively, intersecting at $H$. Draw $CG$ with $G$ on $BD$ and $\angle GCB = 60^\circ$. Show $E$ is the incenter of triangle $BDH$.
4. Mark $K$ on $AC$ such that $\angle KBC = 20^\circ$. Draw $KB$ and $KE$. Show $BE=BC=BK=KE=KD$.
5. (Maria Gelband) Reflect $E$ through $AC$ to point $H$. Show D is on the circumcircle of triangle $BEH$.
6. (Sergei Saprikin) Let the bisector of $\angle ABC$ intersect $AC$ at point $T$. Show $D$ is an excenter of triangle $BET$.
7. (Alexey Borodin) Let $O$ be the circumcenter of triangle $DEC$. Show $BD$ is the perpendicular bisector of $EO$.
8. (Alexander Kornienko) Reflect triangle $ABC$ through $AB$ to triangle $ABC^\prime$ and also relect it through $AC$ to triangle $ACB^\prime$. Show that $C^\prime$, $E$, and $D$ are collinear.