Midpoint theorem (conics)

The midpoint theorem describes a property of parallel chords in a conic. It states that the midpoints of parallel chords in a conic are located on a common line.

The common line (segment) for the midpoints is also called the diameter of a conic. For a circle, ellipse or hyperbola the diameter goes through its center. For a parabola the diameter is always perpendicular to its directrix and for a pair of intersecting lines the diameter goes through the point of intersection.

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  hyperbola

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  circle

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  ellipse

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  parabola

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  intersecting lines

References

• David A. Brannan, David Alexander Brannan, Matthew F. Esplen, Jeremy J. Gray: Geometry. Cambridge University Press, 1999, ISBN 9780521597876, pp. 59–66
• Aleksander Simonic: "On a Problem Concerning Two Conics". In: Crux Mathematicorum, volume 38, no. 9, November 2012, pp. 372–377
• C. G. Gibson: Elementary Euclidean Geometry: An Introduction. Cambridge University Press, 2003, ISBN 9780521834483, pp. 65–68

External links

• Locus of Midpoints of Parallel Chords of Central Conic passes through Center at the Proof Wiki
• midpoints of parallel chords in conics lie on a common line - interactive illustration