Question 78
Let $ABC$ be a triangle with $\angle C = 90^{\circ}$ and $D \in CA$, $E \in CB$, and $k_1, k_2, k_3, k_4$ semicircles with diameters $CA$, $CB$, $CD$, $CE$ respectively, which have common part with the triangle $ABC$. Let also,
$$
k_1 \cap k_2 = \{C, K\},\quad k_3 \cap k_4 = \{C, M\},\quad k_2 \cap k_3 = \{C, L\},\quad k_1 \cap k_4 = \{C, N\}
$$
Prove that $K, L, M$ and $N$ are cocyclic points.
$$
k_1 \cap k_2 = \{C, K\},\quad k_3 \cap k_4 = \{C, M\},\quad k_2 \cap k_3 = \{C, L\},\quad k_1 \cap k_4 = \{C, N\}
$$
Prove that $K, L, M$ and $N$ are cocyclic points.