# What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more equal angles. Further all angles should be less than $\pi/2$. Has anybody optimized this old problem of geometry-teachers?

-
Related to this question is the observation that even if you succeed to draw a "general triangle" as described above, you can still "prove" that it has to equal sides (hence two equal angles). Of course, you cheat somewhere, but it is very subtle. This was shown to me by my math teacher when I was 12, and I never forget the argument. This teacher claimed that "Geometry is the art of making correct reasoning from wrong pictures"; in French "la Géométrie est l'art de raisonner juste sur des figures fausses". –  Denis Serre Apr 25 '13 at 14:15
The moment you have optimized the triangle, it ceases to be general. –  Lee Mosher Apr 25 '13 at 14:58
Well, if you maximise the differences you mention, the answer is 45-60-75. But in the attempt to generate the most general triangle we have arrived at one that is very special... –  Carl Apr 25 '13 at 14:58
Thanks for comments and answers. As Wille says: The student must not identify the general triangle as a special one. So it has to be drawn very unobtrusively. The next question would be then how to draw this triangle freehands. –  Rhett Butler Apr 25 '13 at 19:53
Discussion at tea.mathoverflow.net/discussion/1582/… –  Daniel Moskovich May 3 '13 at 10:10

The book by "Humor in der Mathematik" by Friedrich Wille (from the 1970s or 1980s) contains the tongue-in-cheek theorem "Up to similarity, there is a unique general triangle". (Google book search for "Friedrich Wille" "Allgemeines Dreieck")

"General" is defined as "all angles must differ from each other, and from 90 degress, by at least 15 degrees".

Given some further axioms (like: diagonals of acute angles have to be visually distinct from line of symmetry) it is also shown that there is a unique general quadrilateral, and that this quadrilateral is "teeming" (another technical term) with general triangles.

-
There is a formal, machine-verified proof of this theorem at afp.sourceforge.net/entries/General-Triangle.shtml –  Joachim Breitner May 13 at 12:08
If the largest angle is $\le 75$ (Conditions 1, 3), the next one is $\le 60$ (Condition 2) and the next one is $\le 45$ (Condition 2), then the total angle is $\le 180$ which is impossible on a sphere. –  Mark Sapir May 3 '13 at 21:34