The Swiss mathematician Arnold Droz-Farny published in 1899, without a proof, the statement on a property of a triangle in the Euclidean plane geometry that became known in mathematical literature as the Droz-Farny Line Theorem: Two perpendicular lines through the orthocenter of a triangle intercept on the sides of the triangle three segments whose midpoints are collinear. In this paper we present the first purely synthetic proof of the basic Droz-Farny theorem as it was published by Aymé in 2004, as well as proofs of some of its generalizations, among which the theorem of Goormaghtigh from 1930 turned out to be the most important one. We also exhibit some interesting results related to the basic theorem. For example, Droz-Farny lines generated in a triangle by various choices of two perpendicular lines through the orthocenter constitute a family of triangles that all share the circumcircle and an inscribed conic, having the Euler line and some characteristic points in common, too. In this way a certain property is established of the type that in geometry is often called a porism. This one is also a special case of the renowned Poncelet porism. Finally, we indicate the essential conclusions of the axiomatic analysis of the theorems of Droz-Farny and Goormaghtigh as published by R. and H. Struve in 2016, from the viewpoint of the geometry of reflections.