Triangles of nearly equal 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) ...

Polygons Whose Vertex Triangles Have Equal Area

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...

Cubics Associated with Triangles of Equal Areas

The locus of a point X for which the cevian triangle of X and that of its isogonal conjugate have equal areas is a cubic that passes through the 1st and 2nd Brocard points. Generalizing from isogonal conjugate to P -isoconjugate yields a cubic Z(U,P ) passing through U ; if X is on Z(U, P ) then the P isoconjugate of X is on Z(U,P ) and this point is collinear with X and U . A generalized equal...

Collision Efficiency of Nearly Equal Cloud Drops

Recomputation of the collision efficiency of nearly equal water drops falling in air gave values which increase as the size ratio approaches uiiity for radii of the bigger drop larger than 6 3 p, and decreased as though tending toward zero for smaller radii. For the bigger radii the new computcd values are consistent with experimental results, but for radii around 40 ,U they are not.