  Markov's theorem classifies the worst irrational numbers with respect to
rational approximation and the indefinite binary quadratic forms whose values
for integer arguments stay farthest away from zero. The main purpose of this
paper is to present a new proof of Markov's theorem using hyperbolic geometry.
The main ingredients are a dictionary to translate between hyperbolic geometry
and algebra/number theory, and some very basic tools borrowed from modern
geometric Teichm\"uller theory. Simple closed geodesics and ideal
triangulations of the modular torus play an important role, and so do the
problems: How far can a straight line crossing a triangle stay away from the
vertices? How far can it stay away from the vertices of the tessellation
generated by the triangle? Definite binary quadratic forms are briefly
discussed in the last section.
