  Tripod configurations of plane curves, formed by certain triples of normal
lines coinciding at a point, were introduced by Tabachnikov, who showed that
$C^2$ closed convex curves possess at least two tripod configurations. Later,
Kao and Wang established the existence of tripod configurations for $C^2$
closed locally convex curves. In this paper we generalize these two results,
answering a conjecture of Tabachnikov on the existence of tripod configurations
for all $C^2$ closed plane curves by proving existence for a generalized notion
of tripod configuration. We then demonstrate the existence of the natural
extensions of these tripod configurations to the spherical and hyperbolic
geometries for a certain class of convex curves, and discuss an analogue of the
problem for regular plane polygons.
