IB Mathematics Applications & Interpretation Question Type 7: Finding the Furthest Site Still in the Region for Voronoi Diagrams Exercises

Three sites are located at $A(1,1)$, $B(4,1)$ and $C(1,4)$ inside the square region defined by $0\le x,y\le 5$.

Find the coordinates of the center and the radius of the circle passing through the three sites $A$, $B$ and $C$. Verify that this circle lies entirely within the square region.

Given three sites at $P(1,3)$, $Q(5,3)$ and $R(3,7)$ in the plane, find the center and radius of the largest empty circle whose center is at a vertex of the Voronoi diagram for these sites.

Given three sites at $P(2,2)$, $Q(8,2)$ and $R(5,8)$ inside the region $0\le x,y\le 10$, find the center and radius of the largest empty circle that contains no sites and lies entirely within the square.

Three sites are located at $A(2,2)$, $B(6,2)$ and $C(4,6)$. Find the center and radius of the largest empty circle whose center lies within the triangle formed by these sites.

Four sites are at the corners of the unit square: $(0,0)$, $(1,0)$, $(1,1)$ and $(0,1)$. Determine the largest empty circle inside the square that contains no sites.

Four interior sites at $(1,1)$, $(4,1)$, $(1,4)$ and $(4,4)$ lie in the square $0\le x,y\le 5$. Find the largest circle inside the square containing no sites.

Six sites at $(1,1),(3,1),(5,1),(1,5),(3,5),(5,5)$ lie inside the square $0\le x,y\le 6$. Determine the center(s) and radius of the largest empty circle that contains none of these sites.

A rectangle has vertices at $(0,0)$, $(8,0)$, $(8,3)$ and $(0,3)$. Within this rectangle, there are no interior sites. What is the largest empty circle contained in the rectangle?

Six sites lie in two rows within the rectangle $0\le x\le 8$, $0\le y\le 5$: $(1,1)$, $(4,1)$, $(7,1)$, $(1,4)$, $(4,4)$ and $(7,4)$. Find the center and radius of the largest empty circle in this region that contains no sites.

Explain why, for any set of $n$ sites inside a convex region, the center of the largest empty circle must lie at a vertex of the Voronoi diagram of the sites.

Four sites are located at the vertices of a $6\times 5$ rectangle: $A(0,0)$, $B(6,0)$, $C(6,5)$, $D(0,5)$. Within the convex hull of these four points, determine the center and radius of the largest empty circle that contains no sites.