# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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## A lower bound on the angles of triangles constructed by bisecting the longest sideHTML articles powered by AMS MathViewer

by Ivo G. Rosenberg and Frank Stenger
Math. Comp. 29 (1975), 390-395 Request permission

## Abstract:

Let $\Delta {A^1}{A^2}{A^3}$ be a triangle with vertices at ${A^1},{A^2}$ and ${A^3}$. The process of "bisecting $\Delta {A^1}{A^2}{A^3}$" is defined as follows. We first locate the longest edge, ${A^i}{A^{i + 1}}$ of $\Delta {A^1}{A^2}{A^3}$ where ${A^{i + 3}} = {A^i}$, set $D = ({A^i} + {A^{i + 1}})/2$, and then define two new triangles, $\Delta {A^i}D{A^{i + 2}}$ and $\Delta D{A^{i + 1}}{A^{i + 2}}$. Let ${\Delta _{00}}$ be a given triangle, with smallest interior angle $\alpha > 0$. Bisect ${\Delta _{00}}$ into two new triangles, ${\Delta _{1i}},i = 1,2$. Next, bisect each triangle ${\Delta _{1i}}$, to form four new triangles ${\Delta _{2i}},i = 1,2,3,4$, and so on, to form an infinite sequence T of triangles. It is shown that if $\Delta \in T$, and $\theta$ is any interior angle of $\Delta$, then $\theta \geqslant \alpha /2$.
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