Nice idea but it seems not to always exist:
[![ConvexTriangulation][1]][1]
Vertex $4$'s star is reflex at $3$, vertex $3$'s star is reflex at $4$.
[![ConvexTriangulation][1]][1]
Vertex $4$'s star is reflex at $3$, vertex $3$'s star is reflex at $4$.
Joseph O'Rourke
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