Circle and middles

[ins-]

Let k is the circumference of the triangle ABC. K is the middle of the arc AB (not containing C). L and M are the middles of the sides AC and BC. N and P are the intersection points of KL and LM with k. Prove that L, M, N, P are concyclic.

[nnosipov]

У Вас опечатка: вместо $LM$ должно быть $KM$.

[ins-]

You are correct. I'm sorry for that. The problem can be generalized. Any ideas how can it be solved?

[ins-]

I don't know if that helps to solve the problem - the statement is true not only if L and M are middles of AC and BC. It is true for points L and M such that AL/LC=BM/MC.

[ins-]

I'm sorry it is an easy 8-th grade problem. Don't loose your time.