Interpretation of complex trilinear coordinates The point $X_{5374}$ in the Encyclopedia of Triangle Centres has trilinear coordinates
$$\sqrt{\cot A}:\sqrt{\cot B}:\sqrt{\cot C}$$
If the reference triangle is obtuse, one (and only one) of these coordinates becomes a non-real complex number by virtue of taking the square root of a negative number. There are a few other ETC points that may have complex coordinates, like $X_{5000}$; the new triangle centre I asked about with trilinears $\frac1{\sqrt{a\cos A}}:\frac1{\sqrt{b\cos B}}:\frac1{\sqrt{c\cos C}}$ falls in the same category.
Trilinear coordinates, by definition, are the ratio of directed distances from a point to the triangle edges, but this interpretation only works if all coordinates are real. If a point has complex trilinear coordinates, how can I interpret it? A geometric interpretation would be preferred.
 A: This answer will be a brief and superficial survey of references with interpretations of "imaginary" entities in geometry. When doing Euclidean plane geometry you are implicitly working in the complex Euclidean plane.  In the real plane, a line and circle may not intersect, but in the complex plane they will. You can work purely algebraically, but often there will be geometric interpretations.  For example, the intersection of a disjoint circle and line will yield a conjugate pair of imaginary points, but the line through them will be a "real" line that is in fact the original line.  The two imaginary points of intersection of two disjoint circles will define a real line that is the radical axis of the two circles (see this answer).  And the four imaginary points of intersection of two conics will define two lines that generalize the radical axis and also correspond to a degenerate member of the pencil defined by the two conics.
Back to trilinears, Chapter IX of Whitworth's Trilinear Coordinates, 1866 discusses imaginary points and lines.  The URL is set up with a search term that lets you browse through references to imaginary entities.
Synthetic geometers back in the 19th century also had ways of interpreting the imaginary.  Some of this goes back to Poncelet's "Principle of Continuity" which argued that, for example, the intersection of a conic and line didn't abruptly disappear when they became disjoint. For example, a conic a determines on every straight line an involution by means of pairs of collinear conjugate points and when the double points of this involution are real, they are the points of intersection of the line and conic.  By analogy, involutions where the double points are off the line represent pairs of imaginary points.
It seems that von Staudt pushed these ideas the furthest. See Hatton's Theory of the Imaginary in Geometry for much more detail.  Also the last part of Coolidge's Geometry Of The Complex Domain.
Hamilton and Kettle's Graphs and imaginaries may be worth a glance.
I suspect that this may not directly answer your question of how to interpret a complex trilinear coordinate, but these texts from the past at least give an idea of how geometers have tried to work with imaginaries.  Working out how to do it in synthetic geometry was an impressive achievement but probably less fruitful than analytic methods in forging ahead, so this is perhaps an all but forgotten branch of mathematics.
