Parallel Chords in Ellipse
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The applet illustrates the Euclidean construction of a center of an ellipse by ruler and compass. (The blue lines can be dragged by their end- and midpoints. Just try!)
Cross the ellipse by two parallel lines AB and CD, with points A, B, C, D on the ellipse. Find the midpoints M and N of the segments AB and CD. Line MN is incident to the center of the ellipse. Therefore by choosing a pair of parallel lines with a different direction, the center of the ellipse is found at the intersection of the two midlines.
The construction works for ellipse just because it works for a circle. Indeed, an ellipse is a projection of a circle along one of its axes. Such a transformation maps straight lines on straight lines, preserves line and point incidence and relative lengths of segments on the same line; so that it also preserves midpoints of line segments.
This same construction works for hyperbola as well and, in a sense, for parabola. Except that, for parabola, the center lies at infinity and, as a result, line MN is parallel to the axis of parabola - another line that goes through the same point at infinity.
Copyright © 1996-2018 Alexander Bogomolny