Nice idea but it seems not to always exist:
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[![ConvexTriangulation][1]][1]
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Point $4$'s star is reflex at $3$,
Point $3$'s star is reflex at $4$.
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Here's an argument that those are the only two triangulations.
Each interior point ($3$ and $4$) must be degree-$3$ or
degree-$4$: degree-$3$ to span more than $180^\circ$,
and at most degree-$4$ because there are only four other points.
In the left figure point $3$ has degree-$3$ and point $4$ degree-$4$,
and in the right figure point $3$ has degree-$4$ and point $4$ degree-$3$.
There must be a total of $9$ edges, $6$ of them diagonals. 
Then the diagonals in the two figures are forced.


  [1]: https://i.sstatic.net/SPHxZ.jpg