George Tsintsifas

Let ABC be a triangle and M an interior point. We denoted by R_M the sum of the distances of the point M from the vertices of ABC and by r_M the sum of the distances of the point M from the sides. We found the inequalities between R_M and r_M for M=O,G,I,H that is the pericenter the centroid the incenter and the orthocenter respectively.

It is well known from Mechanics that the polar moment of inertia of a system of weighted points is minimum about the centroid. This can be expressed geometrically and very interesting results can be obtained.

A classification of a conic through three no colinear given points, according the position of its center. Nine point ellipse.

In this note we study the maximal inscribed and the minimal circumscribed ellipse of a cenrally symmetrc convex figure F and we prove two theorems between the area and the remarcable elements of the figure.

The convex figure K has conjugate diameters if and only if its symmetroid K* is an affine image of a Radon curve

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.