  This paper introduces path triangulation of points in a bounded, simply
connected surface region, replacing ordinary triangles in a Delaunay
triangulation with path triangles from homotopy theory. A {\bf path triangle}
has a border that is a sequence of paths $h:I\to X, I=[0,1]$. The main results
in this paper are that (1) a cone $D\times I$ collapses to a path triangle
$h\bigtriangleup K$, extending E.C. Zeeman's collapsible dunce hat cone
theorem, (2) an ordinary path triangle with geometrically realized straight
edges generalizes Veech's billiard triangle, (3) a billiard ball $K\times I$
collapses to a round path triangle geometrically realized as a triangle with
curviliear edges, (4) a geometrically realized homotopy system defined in terms
of free group presentations of path triangulations of finite cell complexes
extends J.H.C. Whitehead's homotopy system geometric realization theorem and
(5) every path triangulation of a cell complex is a good cover.
