Is this point regarded as a triangle center from ETC standard?

Question

Given a reference triangle ABC, create the cevian triangle A’B‘C’ of the symmedian point K of ABC. Then what is the point K called when the reference triangle is A’B‘C’? I search the ETC (https://faculty.evansville.edu/ck6/encyclopedia/Search_13_6_9.html) but find no coordinate relevant to this point. Is it regarded as a triangle center from ETC (Kimberling, C.) standard at all?

Answer 1

In general, that point cannot be "a" triangle center.

Start from a triangle $ABC$ with sides $(a,b,c) = (6,9,13)$. I have located two points $K_1 (0.405026,0.425623,0.169351)$ and $K_2 (0.325602,0.202279,0.472119)^{\color{blue}{[1]}}$ inside $\triangle ABC$:

For each $K_i$, one can construct another triangle $A_iB_iC_i$ so that $K_i$ is the symmedian point of $\triangle A_iB_iC_i$ and at the same time $\triangle ABC$ is the corresponding cevian triangle.

This means the description "$\triangle ABC$ is the cevian triangle of symmedian point $K$ of another triangle" cannot uniquely pin down the point $K$ nor the other triangle.

I have also computed the first trilinear coordinate for $K_1, K_2$ as in the test. They are $\sim 3.1948913$ and $2.568380$ and both of them don't match any numbers of the test page the numbers of first 3000 centers${}^{\color{blue}{[2]}}$ from a local copy I have.

Notes

• $\color{blue}{[1]}$ The 3 numbers after a point are barycentric coordinates with respect to $\triangle ABC$.

• $\color{blue}{[2]}$ The test page in question are giving first coordinates evaluated at $(a,b,c) = (13,6,9)$. The local copy I have is from an older test page evaluated at $(a,b,c) = (6,9,13)$.