Johnson's theorem states that if three equal circles mutually intersect one another in a single point, then the circle passing through their other three pairwise points of intersection is congruent to the original three circles. If the pairwise intersections are taken as the vertices of a reference triangle Δ A B C, then the Johnson circles that are congruent to the circumcircle of Δ A B C have centers J_A | = | -a b c S_A :c(S^2 + S_A + S_C):b(S^2 + S_A S_B) J_B | = | c(S^2 + S_B S_C): - a b c S_B :a(S^2 + S_A S_B) J_C | = | b(S^2 + S_B S_C):a(S^2 + S_A S_C): - a b c S_C, where S, S_A, S_B, and S_C are Conway triangle notation.

anticomplementary circle | anticomplementary triangle | Johnson midpoint | Johnson's theorem | Johnson triangle | Johnson-Yff circles | Yff circles