George Tsintsifas

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.