# Parallel Chords in Ellipse

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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 along one of its axes. Such a transformation maps straight lines on straight lines, preserves line and point incidence and relative lengths of segments on the same line; so that it also preserves 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

- 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

### Pascal and Brianchon Theorems

- Pascal's Theorem
- Pascal in Ellipse
- Pascal's Theorem, Homogeneous Coordinates
- Projective Proof of Pascal's Theorem
- Pascal Lines: Steiner and Kirkman Theorems
- Brianchon's theorem
- Brianchon in Ellipse
- The Mirror Property of Altitudes via Pascal's Hexagram
- Pappus' Theorem
- Pencils of Cubics
- Three Tangents, Three Chords in Ellipse
- MacLaurin's Construction of Conics
- Pascal in a Cyclic Quadrilateral
- Parallel Chords
- Parallel Chords in Ellipse
- Construction of Conics from Pascal's Theorem
- Pascal: Necessary and Sufficient
- Diameters and Chords
- Chasing Angles in Pascal's Hexagon
- Two Triangles Inscribed in a Conic
- Two Triangles Inscribed in a Conic - with Solution
- Two Pascals Merge into One
- Surprise: Right Angle in Circle

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