To find the center of an ellipse; by projective transformation since the
perpendicular bisectors of chords contain the center, but
perpendicularity is not preserved under projectivity, however,
parallelism is; the line through the midpoints of two parallel chords
contain the center of the ellipse. Likewise for the hyperbola.
(Travis Waddington, 1997, now a student at Cal. Tech.)
By symmetry since a circle with center the center of the ellipse that
intersects the ellipse at a point not the endpoint of an axis will
intersect the ellipse at 3 other points, the perpendicular bisector of
any two consecutive intersection points is an axis. This finds the major
and minor axis, then by using the Pythagorean relationship for an
ellipse the foci can be constructed. (Edward McCullough, 1998, now at
Princeton) This will find the axes for the hyperbola, but will not find
the foci.
Using Pascal's theorem with five points on the conic, a tangent can be
constructed to the conic from a given external point using only lines.
(Wang, 1998, now at Harvard) He constructed the tangent, made a macro,
then applied it for the hyperbola to the center. These are the
asymptotes. Having the asymptotes, then the perpendiculars to the axis
at the intersection points (vertices) finds the corners of the
fundamental rectangle. Using again the Pythagorean relationship, the
focal length can be cosntructed, and used to find the foci.
Once the center and the foci are found, then the conics are known.>
There is a book by Salmon on Conic Sections that contains many
constructions. I believe it was published about 1869 and republished
about 5 times. The last edition was about 1920.
It was reprinted in 1954.
Salmon, George: A Treatise of Conic Sections: An Account of Some of the
Most Important Modern Algebraic & Geometric Methods, New York, Chelsea
Publishing Company, 1954.
I have written an article for publication in a high school journal with
the results described above.
Michael Keyton
St. Mark's School of Texas
Dallas, TX 75230
> Date: Tue, 16 Nov 1999 10:10:39 -0600 ()
> From: Clark Kimberling <ck6@evansville.edu>
> To: xpolakis@otenet.gr, yiu@fau.edu, geometry-college@forum.swarthmore.edu
> Subject: conics and dogs
>
> Friends
>
> Suppose you have a conic on which you can place many points. However, you
> are not allowed to "use" any point of intersection of the conic with
> another curve. How can you construct the center, foci, and other features
> of the conic?
>
> This question emerges from The Geometer's Sketchpad, which does a fine job
> of graphing conics using its Locus command. As suggested by paragraph 1,
> you can put points on the conic, and you can see where lines meets the
> conic, but Sketchpad can't use such intersection-points in subsequent
> constructing.
>
> If the answer to this is known, then there's another question: is there a
> large collection of basic Euclidean constructions involving conics
> published under one cover?
>
>
> Clark Kimberling