Nice idea but it seems not to always exist:

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          [![ConvexTriangulation][1]][1]
          ^{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-$3$. After these choices, the diagonals are forced.

Nice idea but it seems not to always exist:

---

          [![ConvexTriangulation][1]][1]
          ^{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.

Nice idea but it seems not to always exist:

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          [![ConvexTriangulation][1]][1]
          ^{Vertex $4$'s star is reflex at $3$, vertex $3$'s star is reflex at $4$.}

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Nice idea but it seems not to always exist:

---

          [![ConvexTriangulation][1]][1]
          ^{Point $4$'s star is reflex at $3$, Point $3$'s star is reflex at $4$.}

---

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-$3$. After these choices, the diagonals are forced.