Loading ...
Sorry, an error occurred while loading the content.
 

Concurrency of four segments joining four points and their isogonal conjugates.

Expand Messages
  • xaraxconstant
    Let us call, for reasons discussed shortly (*), a simple quadrilateral that has its opposite sides (or their extensions) going through two of the vertices of a
    Message 1 of 2 , May 2, 2007
      Let us call, for reasons discussed shortly (*), a simple
      quadrilateral that has its opposite sides (or their extensions) going
      through two of the vertices of a triangle ABC and its opposite
      vertices on isogonal lines through the third vertex, a "perspective
      parallelogram".

      The four segments, joining the four vertices of a "perspective
      parallelogram" with their four isogonal conjugates, are concurrent.



      (*) If we perceive the three vertices of a triangle as the three
      vanishing points of a three point perspective drawing (i.e., perceive
      the three sides of the triangle as the three horizon lines of that
      drawing), the statement of the message becomes self-evident. For a
      brief discussion on the application of this view on similar problems,
      see message 15247.
    • xaraxconstant
      We now consider the general case, of any pair of “perspective parallelograms”, i.e. of any pair of simple quadrilaterals that have their opposite sides (or
      Message 2 of 2 , May 2, 2007
        We now consider the general case, of any pair of “perspective
        parallelograms”, i.e. of any pair of simple quadrilaterals that have
        their opposite sides (or their extensions) going through the same two
        vertices of a triangle ABC and their opposite vertices on the same
        isogonal lines through the third vertex.

        The segments, joining the opposite vertices of any pair of
        “perspective parallelograms”, are concurrent.

        As in the special case, if we perceive the three vertices of a
        triangle as the three vanishing points of a three point perspective
        drawing (i.e., perceive the three sides of the triangle as the three
        horizon lines of that drawing), the statement of the message becomes
        self-evident.
      Your message has been successfully submitted and would be delivered to recipients shortly.