Triangle-proportional-intersecting circles, their complementary hyperbola and an equivalent to the Theorem of Morley (Q2367753)

The author considers circles that intersect the three lines of a given triangle ABC such that the sections on these lines are of length a fixed multiple \(\mu\) of the corresponding side of the triangle \((\mu\)- circle). The tangential circles are an extreme example (with \(\mu=0)\). He shows: For every triangle ABC there is a real number \(m>1\) such that ABC possesses 4, 3 or 2 \(\mu\)-circles depending on whether \(\mu<m\), \(\mu=m\) and \(\mu>m\) (Satz 1). In general, the \(\mu\)-circles cannot be constructed by ruler and compasses (Korollar 1). The centres of all these \(\mu\)-circles are located on a hyperbola whenever ABC has no symmetries (for details see Satz 3).