George Tsintsifas

We found an interesting inequality for the simplex, using barycentric coordinates and then some applications in a triangle.

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