2017 USAMO 第 3 题

Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I.$ Ray $AI$ meets $BC$ at $D$ and $\Omega$ again at $M;$ the circle with diameter $DM$ cuts $\Omega$ again at $K.$ Lines $MK$ and $BC$ meet at $S,$ and $N$ is the midpoint of $IS.$ The circumcircles of $\triangle KID$ and $\triangle MAN$ intersect at points $L_1$ and $L.$ Prove that $\Omega$ passes through the midpoint of either $IL_1$ or $IL.$

设$ABC$是一个外接圆$\Omega$和中心$I的不等边三角形。$射线$AI$在$D$处与$BC$相交，在$M处再次与$\Omega$相交；$直径为$DM$的圆在$K处再次与$\Omega$相交。$线$MK$和$BC$在$S处相交，$N$是$IS的中点。$$\三角形KID$和$\三角形的外接圆MAN$相交于点$L_1$和$L.$证明$\Omega$通过$IL_1$或$IL.$ 的中点