  An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union
of N triangles congruent to T, overlapping only at their boundaries. The
triangle T is the "tile'".
  The tile may or may not be similar to ABC . This paper is the first of
several papers, which together seek a complete characterization of the triples
(ABC,N,T) such that ABC can be N -tiled by T . In this paper, we consider the
case in which the tile is similar to ABC, the case in which the tile is a right
triangle, and the case when ABC is equilateral.
  We use (only) techniques from linear algebra and elementary field theory, as
well as elementary geometry and trigonometry.
  Our results (in this paper) are as follows: When the tile is similar to ABC,
we always have "quadratic tilings'" when N is a square. If the tile is similar
to ABC and is not a right triangle, then N is a square. If N is a sum of two
squares, N = e^2 + f^2, then a right triangle with legs e and f can be N -tiled
by a tile similar to ABC ; these tilings are called "biquadratic". If the tile
and ABC are 30-60-90 triangles, then N can also be three times a square. If T
is similar to ABC, these are all the possible triples (ABC, T, N) .
  If the tile is a right triangle, of course it can tile a certain isosceles
triangle when N is twice a square, and in some cases when N is six times a
square.
  Equilateral triangles can be 3-tiled and 6-tiled and hence they can also be
3n^2 and 6n^2 tiled for any n . We also discovered a family of tilings when N
is 3 times a square, which we call the "hexagonal tilings." These tilings
exhaust all the possible triples (ABC, T, N) in case T is a right triangle or
is similar to ABC . Other cases are treated in subsequent papers.
