Lemoine circles | triplicate-ratio circle

Draw lines P_A Q_A, P_B Q_B, and P_C Q_C through the symmedian point K and parallel to the sides of the triangle Δ A B C. The points where the parallel lines intersect the sides of Δ A B C then lie on a circle known as the first Lemoine circle, or sometimes the triplicate-ratio circle. This circle has circle function l = - (b c(b^2 + c^2))/(a^2 + b^2 + c^2)^2, corresponding to Kimberling center X_141, which is the complement of the symmedian point.

cosine circle | Lemoine axis | Lemoine hexagon | symmedian point | Taylor circle | third Lemoine circle | Tucker circles