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1886. On the three-cusped hypocycloid—Addendum*

Published online by Cambridge University Press:  15 September 2017

Abstract

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Type
Mathematical Notes
Copyright
Copyright © Mathematical Association 1946 

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Footnotes

*

I ventured recently to recall to Professor Hadamard the fact that the asymptotes of any pencil of equilateral hyperbolas are also Simson lines of each triangle whose vertices are three base-points of the pencil. This provides an alternative approach to the generation of the hypocycloid treated in his paper in Math. Gazette, 29, 66-7, 1945.

Professor Hadamard thereupon sent me the following very concise proof of the identity of the asymptotes and Simpson lines and of the fact that they envelop a hypocycloid. He has now very kindly given me permission to communicate his letter to the Gazette

The two small footnotes have been inserted by me.

References

These projections are on L, L', respectively, by definition of the Simpson lines.

MM' and mm' being parallel diameters of the two circles.