Post History

Do these major triangle lines have names?

Kimberling's [Encyclopedia of Triangle Centers](https://faculty.evansville.edu/ck6/encyclopedia/ETC.html) lists, among other things, lines on which each centre is found, but usually listing only two points on the line. As a little project I've assembled these triples to find lines which have many centres, which would seem to be a rough measure of how important the line is. I processed the first 10000 centres in the ETC, and found 12 lines with 99+ centres; of those twelve, I found names for eight in MathWorld. Eight of the twelve lines pass through (at least) two of five of the most important centres, and four of those eight are not listed under [triangle lines](https://mathworld.wolfram.com/topics/TriangleLines.html) in MathWorld.

Do the four lines marked ??? have common names?

* X(1) Incentre -- X(2) Centroid: [Nagel line](https://mathworld.wolfram.com/NagelLine.html) (213+ centres)
* X(1) Incentre -- X(3) Circumcentre: **???** (231+ centres)
* X(1) Incentre -- X(4) Orthocentre: **???** (99+ centres)
* X(1) Incentre -- X(6) Symmedian point: **???** (145+ centres)
* X(2) Centroid -- X(3) Circumcentre -- X(4) Orthocentre: [Euler line](https://mathworld.wolfram.com/EulerLine.html) (960+ centres)
* X(2) Centroid -- X(6) Symmedian point: **???** (197+ centres)
* X(3) Circumcentre -- X(6) Symmedian point: [Brocard axis](https://mathworld.wolfram.com/BrocardAxis.html) (493+ centres)
* X(4) Orthocentre -- X(6) Symmedian point: [van Aubel line](https://mathworld.wolfram.com/vanAubelLine.html) (100+ centres)