Iteration of the construction of Cevian triangles

Question

Pick some triangle center. Construct the Cevian triangle. Consider its angles. If you want to iterate the construction (find the same triangle center for the Cevian triangle, and so on), the angles $\phi,\dots$ of the Cevian triangle should be a simple function of the reference triangle, say $\phi=\lambda*\alpha+\mu*\pi$ (and analogous for $\beta,\gamma$), with $\lambda\in\{0,1,-1,2,-2,1/2,-1/2\}$ for starters. The other angle $\psi$ isn't good for iteration, but I threw it in since it also makes for some interesting centers (this obviously is a totally different problem).

I made a screenshot of a table with some obvious center entries. (Sorry, I am lousy at TeX formatting.) If I still had MATHEMATICA, I would fill the blanks myself by simply computing the center values and looking it up in the above link. But I haven't, such are the risks of graduating :-) Can you help me identifying the missing centers? The first missing I "found" myself eons ago but long forgot its official name...

EDIT: Some (partly glaring obvious) additions and corrections, one center definitely is new (y given). ?: No idea how to define those angles for obtuse triangles.

Answer 1

Half an answer, which I only "accidentally" found in the 40000s of the ETC: The part with the central angles $\psi$ is know as the concept "$t$-central angle point" (duuuuh :-) - the connection is $t=2/3*\lambda+4/3$; excluding the one I already mentioned there are (as $t$, center):

(-2,5964)

(4,5963)

(-1,41622)

(3,42621)

(-2/3,42619)

(8/3,42620)

(1/3,42615)

(5/3,42616)

(1/2,42618)

(3/2,42614)

thus this half of the question can be closed (further $\lambda$ might be added ad lib; it would be interesting how the -clearly transcendent- curve of all these centers looks - EDIT: have a look at curve of the usual $6-9-13$ triangle.)