Given a point A outside of a closed strictly convex plane curve γ, there are two tangent segments from A to γ, the left and the right ones, looking from point A. Problem: Does there exist a curve γ such that one can walk around it so that, at all moments, the right tangent segment is smaller than the left one? In other words, does there exist a pair of simple closed curves, γ and Γ, the former ...

Abstract Given any n points in the plane, not all on same line, there exist two non-collinear triples such that ratio of areas triangles they determine, differs from 1 by at most $$O(\log n/n^2)$$ O ( log n / 2 </mm...

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) ...