## Archive for September, 2006

### The lake

Wednesday, September 27th, 2006### A characteristic property of the ellipse.

Wednesday, September 27th, 2006Let (c) be a compact smooth convex curve in E^{2} 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 e_{0} the unit tangent vector at the point M. We suppose that: angle (AM,e_{0})=angle(BM,-e_{0}).

### A characteristic property of the ellipse.

Tuesday, September 26th, 2006Let (k) be a rotund closed convex curve and A,B interior points. We consider, from an external point M the two support lines MM_{1},MM_{2 }and we assume that for every point M, the angles AMM_{1} and BMM_{2} are equal. In this note we answer to the question whether (k) is an ellipse.

### A characteristic property of a convex centrosymmetric curve

Tuesday, September 26th, 2006**Theorem:** If a closed smooth convex curve (c) in E^{2} 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.

### the flower

Tuesday, September 26th, 2006### The lattice point propery of the ellipse

Sunday, September 24th, 2006Let L_{2 }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 c_{0} of diameter 2R=2^{1/2} has the l.p.c.p. So every figure including c_{0} has the l.p.c.p. For example the equilateral triangle including c_{0} 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 L_{3}. 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.

### Thessaloniki

Sunday, September 24th, 2006### The penguin

Saturday, September 23rd, 2006### The lattice point covering property of the triangle.

Saturday, September 23rd, 2006Every trangle including a unit square in an orthogonal lattice has the lattice point covering property.