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In order to explain Euler's notation of the triangle, it is desirable to use a triangle that does not look equilateral or rectangular or obtuse. There are well known ways to draw a parabola, but I personally have never heard whether tricks have come down from famous teachers to draw a "typical" triangle freehand on the blackboard.

A "theoretical" treatment of the general triangle is given in Friedrich Wille's book (in German) Humor in der Mathematik.

See also https://mathoverflow.net/questions/128716/what-is-the-best-general-triangle

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  • $\begingroup$ Interestingly, I've actually seen at least one article about this (in a many-decades old issue of Mathematics Teacher, I think), although I don't remember anything about the article. However, my preference is to draw a scalene obtuse triangle. $\endgroup$ – Dave L Renfro Apr 17 '18 at 11:13
  • $\begingroup$ First I was taken aback. Obtuse typical? But why not? $\endgroup$ – Wilhelm Apr 17 '18 at 12:10
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    $\begingroup$ I've found that students often think of acute triangles only, and thereby overlook certain issues that occur with obtuse triangles, such as the ambiguous case with the law of sines, and the fact that if you circumscribe a circle about an obtuse triangle then the center of the circle lies in the exterior of the triangle, and also the fact that the altitudes of an obtuse triangle intersect in the exterior of the triangle. $\endgroup$ – Dave L Renfro Apr 17 '18 at 13:02
  • $\begingroup$ Why not draw a few triangles -- one obtuse, one right, and one acute -- to emphasize that the notation applies to all, instead of trying to create a "typically atypical example"? $\endgroup$ – Brendan W. Sullivan Apr 17 '18 at 14:24
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How about drawing a straight line, then fixing a point above that line that is not directly above the middle, and take this as the third vertex? This way, the two upper sides will not have the same length; and the chance that one of them has the same length as the straight line you drew at random is very low.

edit: Looking at Thales theorem might help to avoid angles right angles.

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  • $\begingroup$ Fine. Your own invention or inherited from a teacher? $\endgroup$ – Wilhelm Apr 17 '18 at 12:08
  • $\begingroup$ My own idea, although I don't want to rule out the case that some of my teachers might have done it at some point. $\endgroup$ – Dirk Apr 17 '18 at 13:16
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    $\begingroup$ A refinement would be to fix the segment $[AB]$, assumed to be the longest edge of the triangle, and then make a figure showing all forbidden positions for the third point $C$ : it must lie in the intersection of the two open disks with $[AB]$ as a radius, and must avoid the line orthogonal to $(AB)$ passing through the middle of $[AB]$ and the circle of diameter $[AB]$. Up to symmetry, this leaves two available areas, one for obtuse triangles and one for acute ones. The acute area is quite thin, explaining why constructing a scalene acute triangle is difficult. $\endgroup$ – Benoît Kloeckner Apr 17 '18 at 14:19

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