Problem

Source: USAMO 2017 P3, Evan Chen

Tags: USA(J)MO, USAMO, 2017 USAMO, geometry, Hi, xtimmyGgettingflamed



Let ABC be a scalene triangle with circumcircle Ω and incenter I. Ray AI meets ¯BC at D and meets Ω again at M; the circle with diameter ¯DM cuts Ω again at K. Lines MK and BC meet at S, and N is the midpoint of ¯IS. The circumcircles of KID and MAN intersect at points L1 and L2. Prove that Ω passes through the midpoint of either ¯IL1 or ¯IL2. Proposed by Evan Chen