In this work we report recently obtained results on the full geometric product of two oriented points in conformal geometric algebra Cl(4, 1) that models Euclidean space. We analyze its Cl(3, 0) multivector coefficients and their relationships. Moreover, we derive explicit formulas for extracting from the geometric product of two oriented points both the center position of the point pair and the radius vector of the point pair (half of the point to point distance vector). Finally, we introduce a simple direct method to obtain the two standard conformal points from the geometric product of two oriented points.