Looking for information about a triangle center style construction A student of mine asked me about an odd triangle center construction I'd not seen before; I wasn't able to figure out much about it or identify it amongst the many existing ones, so with his permission I thought I'd ask about it here.
(Actually, he requested that I ask this here quite some time ago and I kept forgetting. Mea culpa!)
The construction is the following. Given a triangle $\triangle ABC$ with circumcenter $O$, let $\mathscr{A,B,C}$ be the circles with center $A,B,C$ and radius $\overline{AO},\overline{BO},\overline{CO}$ respectively. Let $a,b,c$ be the extended edges opposite $A,B,C$ respectively.
Now we construct some new points. Let $A'$ be the midpoint of the segment connecting the closest-to-$A$ point of intersection between $\mathscr{B}$ and $c$ and the closest-to-$A$ point of intersection between $\mathscr{C}$ and $b$; define $B'$ and $C'$ similarly.
The point $Z_{ABC}$ my student is interested in is the intersection of the lines $\overleftrightarrow{AA'}$, $\overleftrightarrow{BB'}$, and $\overleftrightarrow{CC'}$ (in the picture below, $A'=L$, $B'=K$, $C'=M$, and $Z_{ABC}=N$). This common point of intersection does not always exist, but does for a large number of triangles; my student was originally interested in a characterization of when this point exists (this was a challenge problem he created for himself), but now is generally interested in what can be said about $Z_{ABC}$ at all.
 A: Converting a comment to an answer ...

This is $X(3302)$ in Kimberling's Encyclopedia of Triangle Centers. More-accurately, your student's point agrees with $X(3302)$ when the former exists; the latter always exists, because it replaces the notion of "closest intersection" (eg, $F$) of point and side-line with something less fragile, namely
$$F:=B+\frac{r}{c}(A−B)$$
(That is, $F$ is the point at circumradius-distance from $B$ in the direction of $A$.) Incidentally, using the "other" intersections of the circles and lines via $$F:=B−\frac{r}{c}(A−B)$$
(note the sign change) yields $X(3300)$.
ETC describes $X(3302)$ and $X(3300)$ merely as the isogonal conjugates of $X(3301)$ and $X(3299)$, respectively. In turn, $X(3301)$ is defined as the intersection of lines joining $X(35)X(371)$ and $X(36)X(372)$, while $X(3299)$ joins the same points in a different way: $X(35)X(372)$ and $X(36)X(371)$. Following that trail further is left as an exercise to the reader, but I'd be curious to know if there's a deeper connection between the referenced points and your student's construction(s). (New challenge problem, perhaps?)
In any case, your student should write to Kimberling (see "Contact the Keeper" on the ETC's Links page) describing those construction(s).
