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Johnson's Theorem

Let three equal circles with centers , , and intersect in a single point and intersect pairwise in the points , , and . Then the circumcircle of the reference triangle is congruent to the original three.

Furthermore, the points , , , and form an orthocentric system.

Here, the original three circles are known as Johnson circles and the triangle formed by their centers is known as the Johnson triangle. Amazingly, the Johnson triangle circumcircle is also congruent to the circumcircle of the reference triangle and centered at the orthocenter .

A "triquetra" is a figure consisting of three circular arcs of equal radius, and has seen extensive use in heraldry (i.e., coats of arms), specifically in the case of the so-called Borromean rings. The term "Triquetra theorem" was used by Mackenzie (1992) to describe Johnson's theorem.

Mackenzie (1992) generalized this theorem to the case where the three circles do not coincide. In this case, they form six intersection points, and if you partition the points into any two groups of three and look at the circumradii of the points in those groups, there is a nice formula relating them to the radii of the triquetra circles. This formula has some pretty geometric consequences (or "porisms"). Ultimately, Johnson's theorem turns out to be closely related to Poncelet's porism.

REFERENCES:

Emch, A. "Remarks on the Foregoing Circle Theorem." Amer. Math. Monthly 23, 162-164, 1916.

Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 18-21, 1976.

Johnson, R. "A Circle Theorem." Amer. Math. Monthly 23, 161-162, 1916.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 75, 1929.

Kimberling, C. "Encyclopedia of Triangle Centers: X(1478)=Center of Johnson-Yff Circle." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1478.

Mackenzie, D. "Triquetras and Porisms." College Math. J. pp. 118-131. March 1992.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 125-126, 1991.

CITE THIS AS:

Weisstein, Eric W. "Johnson's Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/JohnsonsTheorem.html