In particular, $AI$ is diameter, the angles in $E,E^*$ in the two triangles with hypotenuse $AI$ are right angles, and $\hat D:= \widehat=S^*AD$ implies that $AD$ is tangent in $A$ to $\Delta AL^*S^*$. The tangents in $E$ and $E^*=F$ to $(I)$ intersect in $A$. Let $(I)$ be the circle through $D,E,F$, centered at $I$, so $ID=IE=IF$. For reasons that will be transparent soon, we may like to use $E^*=F$ as an alternative notation. Let us restate, and after each construction we make some simple observations outside the statement, so that finally the statement comes with solution. We use the symmetry / the similarity of properties of the points involved in the construction. Doing this feels already like solving only half of the problem. We will solve the problem by introducing only some of the letters, and showing that the two circles $(TAN)$ and $(LAS)$ are tangent in $A$, the common tangent being. But we are here not in a beauty contest, and even more, it is an important step in the mathematical thinking to restrict to the essential, so let us do this.
Such "beauty errors" are considered in chess composition, well, a displaced parallel maybe, as "true errors", and the composer is urged to restate by giving only the essential. Note that in the given case, the points $B,C$ have no role, so we will delete them from the picture. This is the reverse engineering attempt to solve, maybe mirroring exactly the way the problem was "composed". (by uniqueness of the construction), and suddenly the problem is solved without pain.
One can "accept the challenge", and use the introduced properties with all difficulties that may occur giving the "sincere proof", or one ignores the statement, for "difficult points" $X,Y,Z\dots$ in the problem one introduces (apparently with no connection to the problem) own points $X', y', Z', \dots$ possibly in other order, shows a lot of simple facts with $X', y', Z', \dots$, then finally shows (possibly in other order) that $X=X'$, then $Y=Y'$, then $Z=Z'$, then.
Then one reverts the order of introduction of points, and each point is introduced in the most complicated possible manner. Form the easy properties (for instance intersection of lines) one obtains more complicated properties (the intersection of lines lies on one circle). I have often seen such problems in my youth, the "composer" considers a given situation with easily introduced points, that have easy properties and are introduced in an easy order. This insertion is only a psychological hint and survivor guide for potential Mathematics Olympiad problems in the same style. Please skip, if the next few sentences feel annoying. The picture also reveals some points that the problem missed to introduce, although this would have been a good chance to use the alphabet, so the composer really wanted to make a puzzle out of it. Here is the picture of the story, so that the community can see where the composer of the problem placed almost all letters of the alphabet.
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