Question 58
Given a semicircle $(c)$ with diameter $AB$ and center $O$. On $(c)$ we take a point $C$ such that the tangent at $C$ intersects the line $AB$ at the point $E$. The perpendicular line from $C$ to $AB$ intersects the diameter $AB$ at the point $D$. On $(c)$ we take the points $H$, $Z$ such that $CD = CH = CZ$. The line $HZ$ intersects the lines $CO$, $CD$, $AB$ at the points $S$, $I$, $K$ respectively, and the line through $I$ parallel to the line $AB$ intersects the lines $CO$, $CK$ at the points $L$, $M$ respectively. We consider the circumcircle $(k)$ of the triangle $LMD$, which intersects again the lines $AB$, $CK$ at the points $P$, $U$ respectively. Let $(e_1)$, $(e_2)$, $(e_3)$ be the tangents of $(k)$ at the points $L$, $M$, $P$ respectively and $R = (e_1) \cap (e_2)$, $X = (e_2) \cap (e_3)$, $T = (e_1) \cap (e_3)$. Prove that if $Q$ is the center of $(k)$, then the lines $RD$, $TU$, $XS$ pass through the same point, which lies on the line $IQ$.