I am struggling to understand/believe the idea that the center of the largest empty circle or optimal location for a "toxic waste dump" on a Voronoi diagram is on a vertex.
For example, if I have a Voronoi diagram with sites $A=(1,3), B=(6,4), C= (6,1)$ the Toxic Waste Dump problem tells me that the optimal location for a toxic waste dump would be at the vertex $(3.7, 2.5)$.
What I have read of the Toxic Waste Dump problem is that it tells you the optimal place to put a waste dump so that it is
1. furthest from all existing site
2. not closest to any one particular site.
I understand that it is "fair" to place it on the vertex, but I don't see how that maximizes the distance.
In the example above, why shouldn't the waste dump be placed at $(2.5, 8.5)$?
This point is equidistant from $A$ and $B$ (it lies on their edge) and is further from both $A$ and $B$ than the vertex is.
Any help explaining this would be greatly appreciated. Thank you!