Johnson circles

In geometry, a set of Johnson circles comprises three circles of equal radius r sharing one common point of intersection H. In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): the common point H that they all share, and for each of the three pairs of circles one more intersection point (referred here as their 2-wise intersection).

If any two of the circles happen to osculate, they only have H as a common point, and it will then be considered that H be their 2-wise intersection as well; if they should coincide we declare their 2-wise intersection be the point diametrically opposite H. The three 2-wise intersection points define the reference triangle of the figure.

The concept is named after Roger Arthur Johnson.

Property 2 is also clear: for any circle of radius r, and any point P on it, the circle of radius 2r centered at P is tangent to the circle in its point opposite to P; this applies in particular to P = H, giving the anticomplementary circle C. Property 3 in the formulation of the homothety immediately follows; the triangle of points of tangency is known as the anticomplementary triangle.

For properties 4 and 5, first observe that any two of the three Johnson circles are interchanged by the reflection in the line connecting H and their 2-wise intersection (or in their common tangent at H if these points should coincide), and this reflection also interchanges the two vertices of the anticomplementary triangle lying on these circles.

Since such a homothety is a congruence, this gives property 5, and also the Johnson circles theorem since congruent triangles have circumscribed circles of equal radius.

For property 6, it was already established that the perpendicular bisectors of the sides of the anticomplementary triangle all pass through the point H; since that side is parallel to a side of the reference triangle, these perpendicular bisectors are also the altitudes of the reference triangle.

Property 7 follows immediately from property 6 since the homothetic center whose factor is -1 must lie at the midpoint of the circumcenters O of the reference triangle and H of the Johnson triangle; the latter is the orthocenter of the reference triangle, and its nine-point center is known to be that midpoint.

There is also an algebraic proof of the Johnson circles theorem, using a simple vector computation.

Furthermore, under the reflections about the three sides of the reference triangle, its orthocenter H maps to three points on the circumcircle of the reference triangle that form the vertices of the circum-orthic triangle, its circumcenter O maps onto the vertices of the Johnson triangle and its Euler line (line passing through O, N, H) generates three lines that are concurrent at X(110).

The circumconic and the circumcircle share a fourth point, X(110) of the reference triangle.

Finally there are two interesting and documented circumcubics that pass through the six vertices of the reference triangle and its Johnson triangle as well as the circumcenter, the orthocenter and the nine-point center.

The X(i) point notation is the Clark Kimberling ETC classification of triangle centers.