Are All Triangles Isosceles?

Euclid's great synthesis of Greek geometry, The Elements, was for 
centuries regarded as a model of abstract axiomatic reasoning, but 
in the 19th century a close examination of the foundations of 
mathematics led to the realization that Euclid's axiomatic structure
is actually quite deficient in many respects.  In particular, it
never clearly defines some important fundamental concepts such as 
"betweenness" and "continuity".  In retrospect it's clear that 
Euclid's geometry, rather than giving rigorous proofs of abstract 
concepts suggested by roughly drawn figures, actually gave rough
intuitive proofs based on accurately drawn figures.

One well-known illustration of the logical fallacies to which Euclid's
methods are vulnerable (or at least would be vulnerable if we didn't
"cheat" by allowing ourselves to be guided by accurately drawn
figures) is the "proof" that all triangles are isoceles.  Given an
arbitrary triangle ABC, draw the angle bisector of the interior
angle at A, and draw the perpendicular bisector of segment BC at D,
as shown below:

         

If the angle bisector at A and the perpendicular bisector of BC are
parallel, then ABC is isoceles.  On the other hand, if they are not 
parallel, they intersect at a point, which we call P, and we can draw
the perpendiculars from P to AB at E, and to AC at F.  Now, the two
triangles labelled "alpha" in this figure have equal angles and share
a common side, so they are totally equal.  Therefore, PE = PF.  Also,
since D is the midpoint of BC, it's clear that the triangles labelled
"gamma" are equal right traingles, and so PB = PC.  From this it
follows that the triangles labelled "beta" are similar and equal to 
each other, so we have BE+EA = CF+FA, meaning the the triangle ABC
is isoceles.

Of course, if we attempt to accurately construct the points and
lines described in this proof we will discover that the actual
configuration doesn't look like the figure above.  The point P
necessarily falls outside the triangle ABC.  However, if we carry
out the proof on this basis, and if we now assume the points E 
and F also fall outside the triangle, we still conclude that the 
triangle is isoceles.  This too is an incorrect configuration.

The actual configuration of points given by the stated construction
is for the point P to be outside the triangle ABC, and for exactly
one of the points E,F to be between the vertices of the triangle,
as shown below

        

We still have AE=AF, PE=PF, and PB=PC, and it still follows that 
BE=FC, but now we see that even though AE=AF and BE=FC it does not
follow that AB=AC, because while F is between A and C, E is not
between A and B.  This illustrates the importance of "betweeness"
as a concept in geometry.  M. Pasch was among the first to point
out the importance of "axioms of order" to establish the meaning
of this concept, in his "Vorlesungen uber neuere Geometrie" (1882).
Hilbert incorporated them into his "Foundations of Geometry"

Coincidentally there was a lengthy discussion on the internet not
long ago concerning the question of whether Euclid's Elements, 
viewed as a formal axiomatic system, was fundamentally sound and 
logically consistent.  It was mentioned that there are many serious 
flaws in "The Elements", such as the free use of "superposition" 
to establish congruence results (which is based on pure physical 
intuition of moveable material objects and has no hint of 
justification within Euclid's axiom system), the many implicit 
assumptions of continuity (which are not even acknowledged, let 
alone formalized, in "The Elements"), the meaninglessness of most 
of the "definitions" (many of which are never subsequently used), 
and the complete absence of axioms of order and "betweenness", 
thereby allowing numerous false "proofs" such as the one described
above.  

Oddly enough, although the defects in Euclid's "Elements" are 
common knowledge among historians and philosophers of mathematics, 
they seem to be largely unknown to many working mathematicians, who 
continue to regard "The Elements" as a model of axiomatic rigor
(which, arguably, it was never even intended to be, since the Greek
geometers probably considered themselves to be engaged in what we 
today would call physics, rather than axiomatics).