[FOM] Triangles as circles

[Timothy Y. Chow]

On page 90 of the biography "Lewis Carroll in Numberland" by Robin Wilson 
(W. W. Norton, 2008), we read of a debate that took place in Victorian 
England about the merits and demerits of teaching Euclidean geometry.  
Wilson writes:

   Another objection was that the formal study of Euclid failed to 
   encourage independent thinking, requiring too much rote learning, often 
   with no understanding; indeed, the story was told of an Oxford 
   examination candidate who reproduced a proof from Euclid perfectly, 
   except that in his diagram he drew all the triangles as circles.

The story sounds apocryphal, but never mind that.  To me, a student who 
could perform such a feat might well be demonstrating an incredibly *deep* 
understanding of the axiomatic method.

My question to FOM readers is this: To what extent would it be possible to 
"draw all the triangles as circles" in Euclid?  The work of Avigad, Dean, 
and Mumma (which I admit I have not digested) suggests that it might not 
be possible in some cases.  In other cases, however, it might be quite 
instructive to see what happens.

But maybe I'm just rehashing Vaughan Pratt's questions from December about 
spherical geometry?

Tim