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

Let <!-- MATH $\Delta {A^1}{A^2}{A^3}$ --> be a triangle with vertices at <!-- MATH ${A^1},{A^2}$ --> and . The process of "bisecting <!-- MATH $\Delta {A^1}{A^2}{A^3}$ --> " is defined as follows. We first locate the longest edge, <!-- MATH ${A^i}{A^{i + 1}}$ --> of <!-- MATH $\Delta {A^1}{A^2}{A^3}$ --> where <!-- MATH ${A^{i + 3}} = {A^i}$ --> , set <!-- MATH $D = ({A^i} + {A^{i + 1}})/2$ --> , and then define two new triangles, <!-- MATH $\Delta {A^i}D{A^{i + 2}}$ --> and <!-- MATH $\Delta D{A^{i + 1}}{A^{i + 2}}$ --> .