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

Copyright © 1996-2008 Alexander Bogomolny