Parallel Chords in Ellipse

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?

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.

There is an alternative construction that first draws the tangents to the ellipse.

Conic Sections > Ellipse

Pascal and Brianchon Theorems

72192851