Johnson’s Theorem: You Probably Haven’t Heard of This Circle Theorem

Roger Johnson

Roger A. Johnson (1890–1954), an American geometer, discovered this theorem in 1916. I guess the ancient Greek mathematicians missed the bus on this one. Johnson wrote an influential book Modern Geometry — An Elementary Treatise on the Geometry of the Triangle and the Circle. This book is being reprinted by Dover, and it can be purchased at book stores that carry Dover books; but it may need to be ordered. Note that this is a graduate level text dealing with proofs on theorems from triangle geometry.

Johnson’s Theorem: Given three circles of equal radii r that all pass through a common point H, then the circle through their other three intersections has the same radius r.

So, here is a problem for find Johnson’s Circle. This is a system of three circles. Find the common intersection point of the system (i.e. solve the system).

Hopefully, you got H(2, 3). It is customary to use H for the common intersection point of the three circles. Now, use the “other three” intersection points to find Johnson’s circle and graph. Hopefully, you got the following diagram.

I discuss this problem in my book with link below.

Points, Lines, and Conic Sections: A Sequel to College Algebra

Thanks for reading me.