# Show that Delaunay triangulation contains Gabriel graph on a set of euclidean points

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 $$E_{G} \subset E_{D}$$

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