The M-Points Related to the Perfect Circles in Any Triangle ABC as the Next Points Lying on the Generalized Soddy-Line and About “Square Root Angle”

Abstract

The perfect circles and the amicable triangles are the structures based on any reference triangle ABC. Main part of these structures was presented in Montreal during the ICGG 2012. The perfect circles in the triangle ABC are the family of circles beginning at the Fermat-point (r_x = 0), coming through the incircle (r_x = r) and ending on the circumcircle (r_x = R). The centers of these circles lie on the locus (called here as μ-curve), which is continuous and differentiable. The function of μ-curve is up to the present day unknown, however the mentioned family of the circles has many interesting properties, which could help to find the sought function in the future. The M-points are existent in real only for r_x ≤ r. The Soddy-, Eppstein-, Griffith- and Rigby-points have been defined only for the incircle. The perfect circles allowed to generalize them for 0 ≤ r_x ≤ R. The both M-points (M_i and M_o) are the centers of the circles coming through the intersections of three vertical circles. These circles are for r_x = r (Soddy circles) tangent (on the sides a, b and c of the triangle ABC) and for r_x ≤ r intersect mutually at 6 points (3 inner- and 3 outer-intersections). The circle coming through the inner-intersections will be called M_i-circle and the outer – M_o-circle. The both centers of these circles are so M_i-center and M_o-center. They have many very interesting properties similar to the points, which also lie on the generalized Soddy-line. There appear two new circles, pedal points, their mutual relations and proportions. We also managed to define several derived points, including the vertices of two so-called “square root angles” and a point with a maximum value of a certain proportion.