A beautiful one ;-)

[ins-]

Let k is the circumference of the triangle ABC. The middle of the arc AB, not containing C is P. L and M are the middles of the sides AC and BC. The line through P and L intersects AB and k at the points X and Y. The line through P and M intersects AB and k at the points Z and T. Prove that X, Y, Z and T are concyclic.