Projective Collinearity in a Quadrilateral:
What Is This About?
A Mathematical Droodle

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Explanation

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2017 Alexander Bogomolny

Projective Collinearity in a Quadrilateral


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Points K, L, M, N are selected arbitrarily on the sides AB, BC, CD, and AD of a quadrilateral ABCD. Let P denote the intersection of the diagonals AC and BD and extend the lines PK, PL, PM, PN beyond P to their intersection with the opposite sides. Denote the points of intersection K', L', M', and N', so that the lines KK', LL', MM', and NN' all meet in P. Finally, let E be the intersection of KM and LN and E' be the intersection of K'M' and L'N'. Then the three points E, P, and E' are collinear.

Proof

Introduce two new points: X the intersection of a pair of opposite sides AB and CD and Y the intersection of the other pair of opposite sides BC and AD. If the side lines in either pair are parallel then the corresponding point lies at infinity. If both X and Y lie at infinity then AB||CD and BC||AD, so that the quadrilateral ABCD is parallelogram and P is its center of symmetry: K' is symmetric with K, L' with L, and so on. As an important consequence, E and E' are also symmetric in P. The assertion that E, P, and E' is quite obvious in this case.

If either X or Y (or both) is finite, a projective transformation could be used to move the line XY to infinity. Then the image of ABCD will become a parallelogram, the images of E, P, and E' will be found to be collinear, which might only be possible if the points E, P, E' were collinear in the first place.

References

  1. A. A. Zaslavsky, The Orthodiagonal Mapping of Quadrilaterals, Kvant, n 4, 1998, pp 43-44 (in Russian), pdf is available at https://kvant.mccme.ru/1998/04/.

Related material
Read more...

Bicentric Quadrilateral

  • Collinearity in Bicentric Quadrilaterals
  • Easy Construction of Bicentric Quadrilateral
  • Easy Construction of Bicentric Quadrilateral II
  • Fuss' Theorem
  • Line IO in Bicentric Quadrilaterals
  • Area of a Bicentric Quadrilateral
  • Concyclic Incenters in Bicentric Quadrilateral
  • |Activities| |Contact| |Front page| |Contents| |Geometry|

    Copyright © 1996-2017 Alexander Bogomolny

     62634464

    Search by google: