Let $P \in \mathbb{R} \times \mathbb{R}$ be a set of points on a euclidean plane.

A Delaunay triangulation of $P$ is a graph $DT(P) = (P, E_{D})$ such that $\forall p, q, r \in P$, the edges $(p, q), (q, r), (r, p) \in E_{D}$ if no other point from $P$ is inside the circle passing through $p, q, r$.

A Gabriel graph of $P$ is a graph $GG(P) = (P, E_{G})$ such that $(p, q) \in E_{G} \iff $ the circle with the diamter $d(p, q)$ passing through $p, q$ contains no other point from $P$.

Prove that $\forall P \in \mathbb{R} \times \mathbb{R}, GG(P) \subset DT(P)$ so that $E_{G} \subset E_{D}$

  • $\begingroup$ Interesting question, but: math.stackexchange.com/questions/4092766/… $\endgroup$ – HEKTO Apr 11 at 22:30
  • $\begingroup$ @HEKTO it is the same question of me and the answer is not satisfying IMO $\endgroup$ – Sophie Roseinsta Apr 12 at 7:59
  • $\begingroup$ Posting the same question on multiple sites isn't allowed here - that's the problem $\endgroup$ – HEKTO Apr 12 at 13:36

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