Month: January 2007

Some Geometric and Analytic Inequalities.

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.

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.