ABCDE is regular pentagon. Prove that the points A, B, C and E are concyclic.

The pentagon is regular.
Consider a regular polygon ABCDE.
Take triangles ABC & ABE
BC=AE (sides of a regular polygon are equal)
AB=AB (common side)
∠ABC=∠EAB  (interior angles of a regular polygon are equal)
Therefore, 
Triangle ABC is congruent to triangle ABE
Therefore,
∠AEB=∠ACB (corresponding parts of congruent triangles are equal)
Since these angles are equal, it proves the theorem "Equal chords of a circle subtend equal angles".
Thus, vertices ABCE lie on a circle and are hence CONCYCLIC.

Hence proved