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Parallel Chords in Ellipse
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. Projective transformations map straight lines on straight lines, preserve line and point incidence and relative lengths of segments of the same line; so that they also preserve 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
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