Isosceles Triangles Equal in Perimeter and Area

Rational Triangles with Equal Area

We consider the set of triangles in the plane with rational sides and a given area A. We show there are infinitely many such triangles for each possible area A. We also show that infinitely many such triangles may be constructed from a given one, all sharing a side of the original triangle, unless the original is equilateral. There are three families of triangles (including the isosceles ones) ...

Billiards in Nearly Isosceles Triangles

We prove that any sufficiently small perturbation of an isosceles triangle has a periodic billiard path. Our proof involves the analysis of certain infinite families of Fourier series that arise in connection with triangular billiards, and reveals some self-similarity phenomena in irrational triangular billiards. Our analysis illustrates the surprising fact that billiards on a triangle near a V...

Polygons Whose Vertex Triangles Have Equal Area

Enclosing Isosceles Triangles and Related Problems ∗

Received (received date) Revised (revised date) Communicated by (Name) Given a set of n points in the plane, we show how to compute various enclosing isosceles triangles where different parameters such as area or perimeter are optimized. We then study a 3-dimensional version of the problem where we enclose a point set with a cone of fixed apex angle α.

Minimal Perimeter Triangles

1. INTRODUCTION. Physics often provides mathematics not only with a problem, but also with the idea of a solution. Physical reasoning has traditionally been a fruitful source of mathematical invention, having played a key role in some of the main discoveries of Archimedes, Bernoulli, and Riemann, among many others. In this article we give a much more modest example of a mathematical result that...