Which of the following refers to two circles whose intersection is exactly one point?

Two circles may intersect in two imaginary points, a single degenerate point, or two distinct points.

The intersections of two circles determine a line known as the radical line. If three circles mutually intersect in a single point, their point of intersection is the intersection of their pairwise radical lines, known as the radical center.

Let two circles of radii

and and centered at

and

intersect in a region shaped like an asymmetric lens. The equations of the two circles are

Combining (1) and (2) gives

Multiplying through and rearranging gives

Solving for results in

The chord connecting the cusps of the lens therefore has half-length

given by plugging back in to obtain

Solving for

and plugging back in to give the entire chord length

then gives

This same formulation applies directly to the sphere-sphere intersection problem.

To find the area of the asymmetric "lens" in which the circles intersect, simply use the formula for the circular segment of radius

and triangular height

twice, one for each half of the "lens." Noting that the heights of the two segment triangles are

The result is

The limiting cases of this expression can be checked to give 0 when

and

when

, as expected.

In order for half the area of two unit disks (

) to overlap, set

in the above equation

and solve numerically, yielding

(OEIS A133741).

If three symmetrically placed equal circles intersect in a single point, as illustrated above, the total area of the three lens-shaped regions formed by the pairwise intersection of circles is given by

Similarly, the total area of the four lens-shaped regions formed by the pairwise intersection of circles is given by

Borromean Rings, Brocard Triangles, Circle-Ellipse Intersection, Circle-Line Intersection, Circular Segment, Circular Triangle, Double Bubble, Goat Problem, Johnson's Theorem, Lens, Lune, Mohammed Sign, Moss's Egg, Radical Center, Radical Line, Reuleaux Triangle, Sphere-Sphere Intersection, Steiner Construction, Triangle Arcs, Triquetra, Venn Diagram, Vesica Piscis Sloane, N. J. A. Sequence A133741 in "The On-Line Encyclopedia of Integer Sequences."

Weisstein, Eric W. "Circle-Circle Intersection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circle-CircleIntersection.html