This other question asked:
In an acute triangle $ABC$ let $A_1$ and $A_2$ be the intersections of the altitude from $A$ and the circle with diameter $BC$, with $A_1$ closer to $A$. $B_1$, $B_2$, $C_1$ and $C_2$ are defined cyclically. $A'$ is the intersection of $B_1C_2$ and $B_2C_1$; $B'$ and $C'$ are defined cyclically. Show that $AA'$, $BB'$ and $CC'$ concur at a point $X$.
I have answered that question, showing that the trilinear coordinates of $X$ are $$X=\frac1{a\sqrt{b^2+c^2-a^2}}:\frac1{b\sqrt{c^2+a^2-b^2}}:\frac1{c\sqrt{a^2+b^2-c^2}}$$ $$=\frac1{\sqrt{a\cos A}}:\frac1{\sqrt{b\cos B}}:\frac1{\sqrt{c\cos C}}$$ where $A,B,C,a,b,c$ have their usual meaning (the angles and sides of the underlying triangle).
However, I could not find this centre in Kimberling's Encyclopedia of Triangle Centres, especially since the centre is a real point only for non-obtuse triangles whereas the reference triangles used for the numeric lookup tables are obtuse. (Even the special instruction that the argument of $x$ be taken if it is a complex number doesn't help.)
Before I submit a request to add this centre to the ETC: is it already in there? If not, what properties does this centre have (any lines/curves it is on, conjugates, etc.)?
From the provided trilinear coordinates above it is easy to see that $X$ is the "trilinear square root" of $X_{92}$ (the square root of $x:y:z$ is $\sqrt x:\sqrt y:\sqrt z$) and the "barycentric square root" of the orthocentre $X_4$, but searching for those terms in the ETC yielded no results either.