George Tsintsifas

Theorem

For the triangle T of a minimum perimeter circumscribed to a convex figure (c) the excircles of T are tangent to (c).

Let Q(z)=z³+a₁z²+a₂z+a₃=0 be a cybic in C and  z₁,z₂,z₃ the roots denoted in the plane by the points A,B,C. The Steiner ellipse in the triangle ABC is denoted by E and F₁, F₂ the foci. Van den Berg’s theorem asserts that the roots of the derivate Q'(z) are the complex numbers defined in C by the points F₁ and F₂.

Let (c) be a compact smooth convex curve in E² and A,B are interior points. The point M is on (c) and we denote e(M) the support line of the point M and e₀ the unit tangent vector at the point M. We suppose that: angle (AM,e₀)=angle(BM,-e₀).

Prove that (c) is an ellipse.

Let (k) be a rotund closed convex curve and A,B interior points. We consider, from an external point M the two support lines MM₁,MM₂ and we assume that for every point M, the angles AMM₁ and BMM₂ are equal. In this note we answer to the question whether (k) is an ellipse.

Theorem: If a closed smooth convex curve (c) in E² and an interior point O have the property that (c) possesses parallel supporting lines at the endpoints of every chord through O, then the (c) is centrosymmetric and O is the center.

Let L₂ be an orthogonal lattice in a plane q and F a figure in q. We define that F has the l.p.c.p.(lattice point covering property) if for every position of F in the plane, the figure F includes at least one lattice point.

It is very simple to see that the circle c₀ of diameter 2R=2^1/2 has the l.p.c.p. So every figure including c₀ has the l.p.c.p. For example the equilateral triangle including c₀ has side equal to 2.449 and obviously has the l.p.c.p. Nevertheless we can prove that an equilateral triangle with side 2.154 has l.p.c.p. That is, the problem is to find the minimum conditions between the elements of the figure F, so that the figure F to be able for the l.p.c.p.

Another example for L₃. A cube of side 3^1/2 has obviously the l.p.c.p. because includes a sphere of diameter 3^1/2. But it has been proved by M. Henk that a cube of side  2^1/2 has the l.p.c.p. Remains open the problem for n-cube. M. Henk and the author solved the problem for the ellipsoid in L_n and here is the solution for l.p.c.p. of the ellipse in the plane.