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 Skip Navigation LinksMath Help > Geometry > Circles, Conic Sections > Ellipse Reflection Property

The Ellipse Reflection Property

This elegant proof comes from website http://www.ies.co.jp/math/java/conics/focus_ellipse/focus_ellipse.html 

If the interior of ellipse is silvered to produce a mirror, rays originated at ellipse's focus are reflected to the other focus of the ellipse.

Proof

Part 1.
Point P is a point on the ellipse, which is defined as the locus of points R such that AR+BR is a constant.
L is a tangent line of the ellipse at point P.
Point Q moves on L.
If Q = P, AQ + BQ = minimum.

Part 2.
There are two point A, B, a line L and a moving point Q on L.  Point A' is the reflection of A about L.
When AQ + BQ = minimum at Q = P,
BQ and QA' form BA', which is a straight line.
Angles made by AP, BP and L are equal to each other.

Internet references

http://www.ies.co.jp/math/java/conics/focus_ellipse/focus_ellipse.html

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Summary of geometrical theorems 

 

The webmaster and author of this Math Help site is Graeme McRae.