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The Concurrence of Triads of Common Tangents to Three Circles

Published online by Cambridge University Press:  03 November 2016

E. H. Neville
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Extract

“If pairs of common tangents be drawn to three circles, and if one triad of common tangents be concurrent, the other triad will also be concurrent.”

This is Casey’s enunciation * of a familiar theorem; Salmon † has the same unqualified assertion, and appends a footnote: “This principle is employed by Steiner in his solution of Malfatti’s problem…. [His] construction is: Inscribe circles in the triangles formed by each side of the triangle and the two adjacent bisectors of angles; these circles having three common tangents meeting in a point will have three other common tangents meeting in a point, and these are common tangents to the circles required.”

Type
Research Article
Information
The Mathematical Gazette , Volume 15 , Issue 208 , July 1930 , pp. 134 - 137
Copyright
Copyright © Mathematical Association 1930

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References

page note 134 * Sequel to Euclid, p. 51 (1st ed. 1881; p. 52 in later editions).

page note 134 † Conic Sections, p. 263 (6th ed. 1879). Fiedler reproduces the enunciation and the footnote in his German version of Salmon, p. 425 (4th ed. 1878; p. 573 of 6th ed. 1903).

page note 134 ‡ Ges. Werke, 1, p. 36. The paper, dated 1826, is reprinted from the first volume of Crelle, which appeared in that year.

page note 134 § Q. J. Math. 1, p. 219 (1857).

page note 134 ∥ Vol. 11, p. 313, Q. 255.

page note 134 ¶ Vol. 13, p. 210.

page note 134 ** Q. 255 is not in the list of Questions still unsolved at the end of vol. 12.

page note 134 †† In that heavily documented compendium, Exercices de Géométrie, prepared by the Institut des Frères des Ecoles Chrétlennes. The reference (p. 326 of the 5th edition), apart from a misprint of 1864 for 1854, which would make Hart the earlier discoverer, is surprisingly careless, for Mannheim in his solution naturally refers to the original question, and there should not be the smallest presumption that a published solution is by the propounder himself. A better justification for the warning to verify every reference would be hard to find.

page note 135 * Circle and Sphere, p. 176 (1916). That this cautious author does not mention Hart’s theorem confirms my estimate of the unreliability of the traditional proofs.

page note 135 † Principles of Geometry, v. 4, p. 67 (1925).