On the non-degeneracy property of the longest-edge trisection of triangles

The longest-edge (LE) trisection of a triangle t is obtained by joining the two equally spaced points of the longest-edge of t with the opposite vertex. In this paper we prove that for any given triangle t   with smallest interior angle τ>0τ>0, if the minimum interior angle of the three triangles obtained by the LE-trisection of t   into three new triangles is denoted by τ1τ1, then τ1⩾τ/c1τ1⩾τ/c1, where c1=π/3arctan(3/5)≈3.1403. Moreover, we show empirical evidence on the non-degeneracy property of the triangular meshes obtained by iterative application of the LE-trisection of triangles. If τnτn denotes the minimum angle of the triangles obtained after n   iterative applications of the LE-trisection, then τn>τ/cτn>τ/c where c is a positive constant independent of n  . An experimental estimate of c≈6.7052025350c≈6.7052025350 is provided.