Three middles and equal segments

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Let k is the circumcircle of ABC. M is the middle of the segment AB. The middle of the arc AC (not containing B) is N. The middle of the arc BC (not containing A) is P. K is the intersection point of NM and k. L is the intersection point of PM and k. If X is the intersection point of AC and LN and Y is the intersection point of BC and KP - prove that AX=BY.