CTK Insights

11 Jun

Parallel Chords in Conics

CutTheKnotMath facebook page serves as a place to post math facts, questions or new problems that do not absolutely fit as comments to existing pages at the Interactive Mathematics Miscellany and Puzzles site. One such question has been posted by Cõ Gẫng Lên:

Points A,B,C,D lie on ellipse such that AB||CD. AC and BD meet in H. AD and BC meet in G. Prove that GH crosses AB and CD in their midpoints, E and F.

Why that is so? A short answer is truly cryptic: Because this is true for circles.

For circles the statement holds because of the symmetry: every diameter (a line through the center) perpendicular to a chord passes through the midpoint of that chord (and vice versa.) Now, an ellipse is an affine image of a circle. Perhaps, more conventionally, ellipse is a projective image of circle. Both affine and projectiive transformations are known to preserve incidence and collinearity. The difference between the two stems from the nature of the objects the two geometries - projective and affine - deal with. The subject of the two-dimensional affine geometry are points and lines in a regular plane. Projective geometry deals with all these but, in addition, includes - for any direction, i.e., a family of parallel lines in a regular plane - a point at infinity and a line at infinity as a collection of such add-on points. In this sense, any two lines in projective geometry intersect - at a point at infinity if they are parallel in affine geometry. But if the point of intersection of two lines is finite and is projected to a finite point, the images of the two lines cross at a finite point.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Angles and distances are not preserved under affine or projective transformations, but ratios of the lengths of segments on the same line are preserved by affine and, if points at infinity are mapped to points at infinity, by projective transformations. (In general, projective transformations preseve the cross-ratio of four collinear points.)

This is is why the statement also holds for parabola and hyperbola, the other two conic sections, all of which are projective images of circle. Under affine and projective transformations, the image of a diameter of a circle may cease to be perpendicular to the image of a chord at hand, but will continue passing through its midpoint and the center of the resulting conic. For two chords (AB and CD that remain parallel, all the incidences and collinearities also remain in place.

References

  1. D. A. Brannan, M. F. Esplen, J. J. Gray, Geometry, Cambridge University Press, 2002
  2. H. Eves, A Survey of Geometry, Allyn and Bacon, Inc., 1972
  3. F. Klein, Elementary Mathematics from an Advanced Standpoint: Geometry, Dover, 2004
  4. D. Pedoe, Geometry: A Comprehensive Course, Dover, 1988

4 Responses to “Parallel Chords in Conics”

  1. 1
    Hubert Shutrick Says:

    This is a property of the complete quadrilateral. The conic is not necessary.

  2. 2
    admin Says:

    Sadly, that's true http://www.cut-the-knot.org/pythagoras/HarmonicRatio.shtml

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