On 06/04/2015 23:52, James Propp wrote:
I still don't see it.
In my picture of a triangle in which "AB is the middle S and BC is the end S", there are two triangles, ABC and ABC', with A, C , and C' collinear, and where B is the apex of isosceles triangle BCC'. The circle centered at B with radius BC does indeed go through C', but I don't see why it goes through A, so I don't see why it's the common circumcircle of the two triangles.
What's common is not the *circumcircle* but the *circumradius*. Two ass-congruent triangles have equal circumradius, so no sequence of ass-congruences can produce similar but incongruent triangles. (No, I don't know why so many other respondents have been saying "same circumcircle" when they mean "same circumradius".) -- g