Midpoint theorem (conics)

In geometry, 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 or line segment for the midpoints is called the diameter. 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 (from a degenerate conic) the diameter goes through the point of intersection.

Gallery (${\displaystyle e}$= eccentricity):

• 
  circle (${\displaystyle e}$=0)
• 
  ellipse (${\displaystyle e}$<1)
• 
  parabola (${\displaystyle e}$=1)
• 
  hyperbola (${\displaystyle e}$>1)
• 
  intersecting lines (${\displaystyle e}$=∞)

• David Alexander Brannan, Matthew F. Esplen, Jeremy J. Gray (1999) Geometry Cambridge University PressISBN 9780521597876, pages 59–66
• Aleksander Simonic (November 2012) "On a Problem Concerning Two Conics", Crux Mathematicorum, volume 38(9): 372–377
• C. G. Gibson (2003) Elementary Euclidean Geometry: An Introduction. Cambridge University PressISBN 9780521834483 pages 65–68

• 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