# Angle Bisectors in Ellipse II

Let A and B be two points on an ellipse with foci E and F. The tangents to the ellipse at A and B meet in S. Prove that

22 January 2016, Created with GeoGebra

### 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

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Copyright © 1996-2018Alexander Bogomolny

Let A and B be two points on an ellipse with foci E and F. The tangents to the ellipse at A and B meet in S. Prove that

### Proof

To see why this is so, draw an ellipse through S *confocal* to the given one. This reminds of the configuration which showed the ellipse as an envelope of a family of straight lines.

So we now have two confocal ellipses: *ellipse*_{1} (the original one) and *ellipse*_{2} (passing through S.) The configuration admits the following interpretation. Draw a tangent to *ellipse*_{1} at A till it hits *ellipse*_{2} in S. Reflect it at S and continue to the next intersection with *ellipse*_{2}, and so on. All so constructed lines will touch an ellipse confocal with *ellipse*_{2} which is bound to be *ellipse*_{1} since the latter, by the construction, is already tangent to the first line AS. It follows that the second line in the chain is necessarily BS, implying that AS and BS are equally inclined to the tangent to *ellipse*_{2} at S. But so are ES and FS, and we are done.

### 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 © 1996-2018Alexander Bogomolny

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