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This paper introduces path triangulation of points in a bounded, simplyconnected surface region, replacing ordinary triangles in a Delaunaytriangulation 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 resultsin 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 conetheorem, (2) an ordinary path triangle with geometrically realized straightedges generalizes Veech's billiard triangle, (3) a billiard ball $K\times I$collapses to a round path triangle geometrically realized as a triangle withcurviliear edges, (4) a geometrically realized homotopy system defined in termsof free group presentations of path triangulations of finite cell complexesextends J.H.C. Whitehead's homotopy system geometric realization theorem and(5) every path triangulation of a cell complex is a good cover.

Path Triangulation, Cycles and Good Covers on Planar Cell Complexes.  Extension of J.H.C. Whitehead's Homotopy System Geometric Realization and  E.C. Zeeman's Collapsible Cone Theorems