Question Type 8: Drawing the largest empty circle Exercises

Four points lie at $P_1(-3,0)$, $P_2(3,0)$, $P_3(0,4)$ and $P_4(0,-4)$. Find the center and radius of the largest empty circle that contains none of these four points in its interior.

Given the four points $A(0,0)$, $B(4,0)$, $C(4,4)$ and $D(0,4)$, find the largest empty circle whose center lies inside the convex hull of these points and contains none of them in its interior.

Given the triangle with vertices $A(0,0)$, $B(4,0)$ and $C(1,3)$, find the center and radius of its circumcircle.

Consider the rectangle with vertices $A(1,1)$, $B(7,1)$, $C(7,5)$ and $D(1,5)$. Determine the center and radius of the largest empty circle that contains none of these four points in its interior.

A regular hexagon has side length $s=2$. Determine the radius of the largest empty circle that contains no vertex in its interior (i.e. he incircle).

Triangle $ABC$ has vertices $A(0,0)$, $B(5,0)$ and $C(1,1)$. Determine the center and radius of the largest empty circle that contains no vertices in its interior.

A regular pentagon has circumradius $R=1$. Find the radius of its inscribed circle (the largest empty circle that contains no vertex in its interior).

Explain why, if all given sites lie on a straight line, there is no finite largest empty circle that excludes them.

Given a triangle with general vertices $A(x_1,y_1)$, $B(x_2,y_2)$ and $C(x_3,y_3)$, derive the coordinates of its circumcenter.

Show that the center of the largest empty circle among a set of sites in the plane always lies at a vertex of the Voronoi diagram of those sites.

Prove that for any triangle $ABC$, the largest empty circle excluding its vertices is \begin{itemize} \item the circumcircle if the triangle is acute, \item the circle with diameter equal to the longest side if the triangle is obtuse. \end{itemize}