Conjugate Diameters in Ellipse
The line joining the midpoints of two parallel chords in an ellipse passes through the center of the ellipse. Moreover, the tangents to the ellipse at the extremities of the chords all meet on the same straight line.
Any chord that passes through the center of an ellipse is call its diameter. It follows that the family of parallel chords define two diameters: one in the direction to which they are all parallel and the other the locus of their midpoints. Such two diameters are called conjugate. The term suggests some symmetry between the two directions, the symmetry that is not apparent from the construction.
What if applet does not run? 
There is indeed a symmetry:
 The midpoints of the chords parallel to one of the conjugate diameters lie on the other.
 The tangents at the end points of the chords parallel to one of the conjugate diameters meet on the other.
 To paraphrase the previous statement: the poles of polars parallel to one of the conjugate diameters lies on the other
 And vice versa: the polars of the poles on one of the conjugate diameters are parallel to the other.
The statement is easily proved analytically if we start with the equation
Thus ellipse is a curve defined by the radiusvector
r(t) = (a cos(t), b sin(t)). 
For a fixed t, we are interested in two points,
r(t + v) = (a (cos(t)cos(v)  sin(t)sin(v)), b (sin(t)cos(v) + cos(t)sin(v))),  
r(t  v) = (a (cos(t)cos(v) + sin(t)sin(v)), b (sin(t)cos(v)  cos(t)sin(v))). 
The slope of the difference, say, r(t + v)  r(t  v) is
(r(t + v) + r(t  v)) / 2 = cos(v) (a cos(t), b sin(t)) 
which is a parameterization (with parameter cos(v)) of the chord with the slope of
Making use of the parameterization
OP² + OQ²  = (a² cos²(t) + b² sin²(t)) + (a² sin²(t) + b² cos²(t))  
= (a² cos²(t) + a² sin²(t)) + (b² sin²(t) + b² cos²(t))  
= a² + b², 
independent of t. This is known as the first theorem of Apollonius: for the conjugate (semi)diameters OP and OQ,
The area of the parallelogram formed by the tangents parallel to a pair of conjugate diameters can be computed via the determinant:

This is known as the second theorem of Apollonius.
References
 G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960
 C. Zwikker, The Advanced Geometry og Plane Curves and Their Applications, Dover, 2005
Conic Sections > Ellipse
 What Is Ellipse?
 Analog device simulation for drawing ellipses
 Angle Bisectors in Ellipse
 Angle Bisectors in Ellipse II
 Between Major and Minor Circles
 Brianchon in Ellipse
 Butterflies in Ellipse
 Concyclic Points of Two Ellipses with Orthogonal Axes
 Conic in Hexagon
 Conjugate Diameters in Ellipse
 Dynamic construction of ellipse and other curves
 Ellipse Between Two Circles
 Ellipse in Arbelos
 Ellipse Touching Sides of Triangle at Midpoints
 Euclidean Construction of Center of Ellipse
 Euclidean Construction of Tangent to Ellipse
 Focal Definition of Ellipse
 Focus and Directrix of Ellipse
 From Foci to a Tangent in Ellipse
 Gergonne in Ellipse
 Pascal in Ellipse
 La Hire's Theorem in Ellipse
 Maximum Perimeter Property of the Incircle
 Optical Property of Ellipse
 Parallel Chords in Ellipse
 Poncelet Porism in Ellipses
 Reflections in Ellipse
 Three Squares and Two Ellipses
 Three Tangents, Three Chords in Ellipse
 Van Schooten's Locus Problem
 Two Circles, Ellipse, and Parallel Lines
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny71409502