It is known that if from a given point perpendiculars be let fall on the four Simson lines formed from the four triangles made by taking every, three of four points concyclic with the first, the feet of these perpendiculars lie on a straight line proposed to be called the Simson line of the quadrangle formed by the four points; it is also known that this process can be extended. I propose to examine various results connected with these lines.