Some Geometric and Analytic Inequalities.

# Month: January 2007

## An inequality in the Cartesian plane.

Let A(1,0), B(-1,0), C(0,1), D(0,-1) be points in the Cartesian plane and P so that OP is no less than 1. Then, it holds:

|AP-BP|+|CP-DP| is no less than 2.

## Inequalities for a simplex and a triangle.

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

## Inequalities 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.