Trilinear imitators in flank triangles: an alternative to triangle centre friendship

The Vecten configuration is formed by erecting external squares on the sides of a triangle ABC, producing three flank triangles. Given a point Q in the plane, we construct for each flank the point having the same trilinear coordinates with respect to that flank as Q has with respect to ABC; denote these corresponding points by QA,QB,QC. We show that the lines AQA,BQB,CQC are concurrent if and only if Q lies on the Thomson cubic, in which case Q is called an imitative point and the concurrency point is called the imitator of Q. Moreover, all such imitators lie on the Darboux cubic. We also relate this to a construction of Montesdeoca which yields a perspectivity between ABC and a derived triangle and produces the same concurrency point, thereby linking the two viewpoints. We note that the imitation framework is conceptually close to van Lamoen's friendship among triangle centres. Analogous relations can be investigated using barycentric (or other) coordinates and in configurations beyond Vecten's. Simple GeoGebra tools are provided for experimentation.