This is a paper on geometry in the usual real Euclidean space. We  treat the distance between two skew lines, by using a very beautiful instance of the parallel sum of two matrices. We display the two points where the shortest distance between two skew lines is achieved. This paper offers the reader a great amount of facts aiming to rise awareness about delicate questions - when dealing with infinite dimensional spaces: on closed sum of subspaces; on the non-linearity of the projector operator; and on the necessity of the sum of (closed) subspaces to be closed in order to guarantee the existence of the Moore-Penrose generalized inverse of the sum of two projectors. In making use of approximation theory results, this text is sprinkled with observations regarding concepts stemming from linear algebra to functional analysis to topology.

distance, Moore-Penrose inverse, best approximation pairs, conjugacy principle.