Generation of parabola via Apollonius' mesh

Apollonius' theorem can be used to generate parabolae as envelopes of the set of their tangent lines.

Assume a parabola with two points A and B and their tangents AS and BS is given. Pick a number n and divide AS and BS into n equal intervals. Label division points on AS with numbers 1, 2, 3, ... counting from S, and mark those on BS counting from B. Connect the points with the same labels. From Apollonius' theorem, the lines will envelope the parabola [Dörrie, pp. 220-222, Wells, p. 171].

If one starts with just two segments AS and BS, the emerging parabola will touch them at points A and B.

References

1. H. Dörrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY, 1965
2. D. Wells, Curious and Interesting Geometry, Penguin Books, 1991 p26,Galileo-catenary

Parabola

• The Parabola
• Archimedes Triangle and Squaring of Parabola
• Focal Properties of Parabola
• Geometric Construction of Roots of Quadratic Equation
• Given Parabola, Find Axis
• Graph and Roots of Quadratic Polynomial
• Greg Markowsky's Problem for Parabola
• Parabola As Envelope of Straight Lines
• Generation of parabola via Apollonius' mesh
• Parabolic Mirror
• Three Parabola Tangents
• Three Points on a Parabola
• Two Tangents to Parabola

Conic Sections

Copyright © 1996-2010 Alexander Bogomolny