Several days recently swirled down the drain in a depression-spiral. Needing a way out, I spent my Saturday morning playing with triangles, after first getting plenty of sleep. It worked. This technique, however, probably would not transfer to those who are not geometry obsessives. Perhaps any favorite activity would work? I leave that to others to explore.
Here’s what I did that worked for me:
The original triangle is ABC, and is in bold black. The bold blue line is its Euler Line, and contains the orthocenter (M), circumcenter (G), nine-point center (K), and centroid (point W). It does not, however, contain the incenter (S).
It struck me as odd that the incenter would be different in this way, so I investigated it further. It is the point of concurrence of the three angle bisectors of a triangle. On a lark, I constructed the midsegments of triangle ABC, forming a new, smaller triangle, shown in red. When I then found the incenter of this smaller triangle (Z), it appeared to be collinear with S and W. I checked; it was, and this line is shown in bold yellow. Moreover, the process could be continued with even smaller midsegment-triangle incenters, and they were also on this yellow line.
I wondered if I had discovered something new, and started to check. It didn’t take long to find out that Nagel had beaten me to it. The Nagel line is the official name of this yellow line I stumbled upon, and here is my source: http://mathworld.wolfram.com/NagelLine.html — but, as far as I know, I did discover that these midsegment-derived points also lie on the Nagel line.
Someone else may have known this before, of coruse. I don’t know, and it doesn’t matter to me, for I had my fun morning playing with triangles, and now feel better than I have in days.
[Side note: this is my 100th post, and I’d like to thank all my readers and followers, and also thank, especially, those who encouraged me to try WordPress to get a fresh start after Tumblr-burnout. It worked!]