Showing points lie on the same circle

Question

Let $\Delta ABC$ be a triangle and consider the line $t$ be tangent to the circumcircle of $\Delta ABC$ at the vertex $C$. The line $p$ parallel to the tangent $t$ intersects the lines $AB$ and $AC$ at the points D and E, respectively.

Prove that the points $A, B, D, E$ belong to the same circle.

Answer 1

• Because $FC||DE$ we have $\angle DEA = \angle FCA$
• Because of tangent-chord theorem we have $\angle ABC = \angle FCA$

Thus $\angle DEA = \angle ABC$ and so $A,B,D,E$ are concyclic.