  We explore the relationship between Brouwer's intuitionistic mathematics and
Euclidean geometry. Brouwer wrote a paper in 1949 called "The contradictority
of elementary geometry". In that paper, he showed that a certain classical
consequence of the parallel postulate implies Markov's principle, which he
found intuitionistically unacceptable. But Euclid's geometry, having served as
a beacon of clear and correct reasoning for two millenia, is not so easily
discarded. Brouwer started from a "theorem" that is not in Euclid, and requires
Markov's principle for its proof. That means that Brouwer's paper did not
address the question whether Euclid's "Elements" really requires Markov's
principle. In this paper we show that there is a coherent theory of
"non-Markovian Euclidean geometry." We show in some detail that our theory is
an adequate formal rendering of (at least) Euclid's Book~I, and suffices to
define geometric arithmetic, thus refining the author's previous investigations
(which include Markov's principle as an axiom). Philosophically, Brouwer's
proof that his version of the parallel postulate implies Markov's principle
could be read just as well as geometric evidence for the truth of Markov's
principle, if one thinks the geometrical "intersection theorem" with which
Brouwer started is geometrically evident.
