Three Tangents, Three Chords in Ellipse

The applet below offers an illustration to a statement which is a particular case of Pascal's Hexagon Theorem: given a triangle T₁T₂T₃ inscribed in a non-degenerate conic. Let t₁, t₂, t₃ be the tangents to the connic at points T₁, T₂, T₃. Then the points of intersection of T_iT_j with t_k (i, j, k all different indices 1, 2, 3) are collinear.

How does Pascal's theorem apply? Let the vertices of an inscribed hexagon coalesce in adjacent pairs producing a triangle with three sides of the hexagon degenerating into the tangents at the vertices of the so obtained triangle.

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
• Conjugate Diameters in Ellipse
• Dynamic construction of ellipse and other curves
• Ellipse in Arbelos
• Ellipse Between Two Circles
• 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

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Copyright © 1996-2012Alexander Bogomolny