A Lower Bound on the Angles of Triangles Constructed by Bisecting the Longest Side

Let AA A A be a triangle with vertices at A1, A2 and A3. The process of "bisecting AA A A is defined as follows. We first locate the longest edge, A'Ai+i of AA 1A2A3 where A'+3 = Al, set D = (A' + Ai+l)/2, and then define two new triangles, AA'DAi+2 and ADAi+1Ai+2. Let Aqq be a given triangle, with smallest interior angle a > 0. Bisect AQ0 into two new triangles, Aj., í = 1, 2. Next, bisect each triangle A,., to form four new triangles Aj,-, í = 1, 2, 3, 4, and so on, to form an infinite sequence T of triangles. It is shown that if A £ T, and 6 is any interior angle of A, then 8 > a/2. Results. Let AABC be a triangle with vertices at A, B and C. The procedure "bisect AABC is defined as follows. We form two triangles from AABC by locating the midpoint of the longest side of AABC and drawing a straight line segment from this midpoint to the vertex of AABC which is opposite the longest side. (If there is more than one side of greatest length, we bisect any one of them.) For example, if BC is the longest side of AABC, we set D = (B + C)/2 to form two new triangles AABD and AADC. Let AABC be a given triangle with interior angles a, ß and y located at A, B and C, respectively. We form an infinite family T(A, B, C) of triangles as follows. We first bisect A0Q = AABC to form two new triangles Alt, i = 1, 2. We next bisect each of these two triangles to form four new triangles A2i, i= 1,2,3,4. Next, we bisect each of these four triangles to form eight new triangles A3/, i = 1, 2, 3, ..., 8, and so on. It is convenient to apply this procedure of bisections in order to refine the mesh in the finite element approximations of solutions of differential equations (see, e.g., [1]). Recently [2], this procedure of bisecting triangles was used to obtain a two-dimensional analogue of the one-dimensional method of bisections for solving nonlinear equations. A criterion of convergence of the above procedures is that the interior angles of An¡ do not go to zero as n —> °°. The Schwarz paradox [3, pp. 373-374] provides an explicit example of a situation in which triangles are used to approximate the area of a cylinder. In this case, the sum of the areas may not converge to the area of the cylinder as the length of each side of the triangles approaches zero, and the number of triangles approaches infinity, if the smallest interior angle of each triangle approaches zero. Received January 2, 1974. AMS (MOS) subject classifications (1970). Primary 50B30, S0B15; Secondary 41A63, 6SN30, 6SH10.