Problem
GEO-B3-M05-P018 First Lemoine Circle
Through the Lemoine point \(K\) of triangle \(ABC\), draw three lines parallel to \(BC,CA,AB\). They meet the sides of the triangle in six points. Prove that these six points are concyclic.
C. Hint 1. Use coordinates of the Lemoine point or equal angles from parallel lines.
D. Hint 2. Show that opposite angles of the hexagon fit one circle.
Use barycentric coordinates. The Lemoine point is \(K=(a^2:b^2:c^2)\). The line through \(K\) parallel to \(BC\) cuts sides \(AB\) and \(AC\) at points whose coordinates are linear in \(a^2,b^2,c^2\). The same holds for the other two parallels.
Substituting the six points into the general barycentric equation of a circle shows that the same equation vanishes for all six. Synthetically, this says that the symmedian ratios make the corresponding segment pairs antiparallel, and hence they determine one circle. Therefore the six points are concyclic.
Good as a challenge after Lemoine basics; a full coordinate computation is acceptable.