Question Type 8: Drawing the largest empty circle Exercises

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.

Triangle $ABC$ has vertices $A(0,0)$, $B(5,0)$ and $C(1,1)$. Determine the center and radius of the smallest circle that contains all three vertices.

Consider the rectangle with vertices $A(1,1)$, $B(7,1)$, $C(7,5)$, and $D(1,5)$. Determine the center and the radius of the largest empty circle whose center lies within the rectangle.

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.

Show that if the center of the largest empty circle among a set of sites lies strictly within the convex hull of the sites, it is a vertex of the Voronoi diagram.

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

A regular pentagon has circumradius $R=1$. Find the exact radius of its inscribed circle.

A regular hexagon has side length $s=2$. Determine the radius of the largest circle contained within the hexagon (i.e. the incircle).

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

Prove that for any triangle $ABC$, the smallest enclosing circle is:

• the circumcircle if the triangle is acute
• the circle with diameter equal to the longest side if the triangle is obtuse