seven-point circle

The Brocard circle, also known as the seven-point circle, is the circle having the line segment connecting the circumcenter O and symmedian point K of a triangle Δ A B C as its diameter (known as the Brocard diameter). This circle also passes through the first and second Brocard points Ω and Ω', respectively. It also passes through Kimberling centers X_i for i = 3, 6, 1083, and 1316. It has circle function l = - (b c)/(a^2 + b^2 + c^2), corresponding to the triangle centroid G and giving trilinear equation a b c(α^2 + β^2 + γ^2) = a^3 βγ + b^3 γα + c^3 αβ (Carr 1970; Kimberling 1998, p. 233).

Brocard angle | Brocard diameter | Brocard line | Brocard points | Brocard triangles | cosine circle