# Hofstadter Triangles and Points

What if applet does not run? |

Did you try checking (say out of curiosity) the small radio buttons at the right bottom portion of the applet? If you did, you might have noticed three families of *concurrent* lines, i.e. the lines that meet at a single point:

- PU, QV, RW
- AP, BQ, CR
- AU, BV, CW

Toying with the applet provides a convincing demonstration that in all three families the lines are indeed concurrent regardless of the value of n. The three points are in general distinct.

There are many families of concurrent lines in a triangle. The best known are the angle bisectors, medians, altitudes, and perpendicular bisectors. There are more. On the Web, Clark Kimberling of University of Evansville has collected a respectable list of such points and the corresponding families of lines. It's there that I also learned, albeit somewhat late, that "Morley's" triangles with 1/3 replaced by a real r have been known for a while as the Hofstadter triangles.

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Copyright © 1996-2017 Alexander Bogomolny

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