Proving the non-degeneracy of the longest-edge trisection by a space of triangular shapes with hyperbolic metric

From an initial triangle, three triangles are obtained joining the two equally spaced points of the longest-edge with the opposite vertex. This construction is the base of the longestedge trisection method. Let D be an arbitrary triangle with minimum angle a. Let D be any triangle generated in the iterated application of the longest-edge trisection. Let a0 be the minimum angle of D. Thus a0 P a=c with c 1⁄4 p=3 arctan ffiffi 3 p =11 ð Þ is proved in this paper. A region of the complex half-plane, endowed with the Poincare hyperbolic metric, is used as the space of triangular shapes. The metric properties of the piecewise-smooth complex dynamic defined by the longest-edge trisection are studied. This allows us to obtain the value c. 2013 Elsevier Inc. All rights reserved.