2 Directions: Reflection Line (dashed) and Guiding Line (red)
This is a generalization of the perpendicularity construction at the Euclidean case: There is also a reflecting direction (dashed lines), but to which a Guiding direction (red lines) was added (in the euclidean case, this guiding line for the reflections is always perpendicular to the reflecting direction). The second reflection is made about the bisectrices of both lines (shown in light blue).
It is possible to see that these (light blue) bisectrices are also the directions of the main axes for the circumscribed ellypse (which, as expected, is centered at the interesection of the three (yellow) "mediatrices").
The (dashed) reflecting direction is also the diagonal of the rectangle made with the tangents at the ellypses vertexes.
2 Directions :Reflection Line (blue) with Guiding Line (green) orthogonal to it, and an arbitrarty "bisecting" direction (dashed line) for the second reflection.
The three (yellow) "mediatrices" meet again at a common point, which is the center of a (blue) ellypse.
This ellypse has its axes aligned with the two reflecting directions (light blue and green lines).
3 Directions: Reflection Line, Guiding Line and General Bisector
This ia a generalization to both previous cases.
No restriction is put for the first reflection and its guiding direction (blue lines), and the second reflection is made about an arbitrary direction (dashed) .
The three (yellow) "mediatrices" meet again at a common point, which is the center of an ellypse passing at the three vertexes.