Euclidean Construction of Center of Ellipse

 

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The applet illustrates the Euclidean construction of a center of an ellipse by ruler and compass.

Choose any two points on Ellipse (A and B in the applet). It takes just a ruler and a drawing of a few lines to find the tangents to the ellipse at A and B. Assume the tangents meet at point S. Let M be the midpoint of AB. As a matter of fact line SM passes through the center of the ellipse. So to find the center one needs to repeat the construction for an additional pair of two points. The two lines SM will meet at the center.

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 SM is parallel to the axis of parabola - another line that goes through the same point at infinity.

There is an alternative construction that draws a line through the midpoints of two parallel chords in ellipse.

Conic Sections > Ellipse

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