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XXIV.—On Fagnani's Theorem

Published online by Cambridge University Press:  17 January 2013

H. F. Talbot
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Extract

Before proceeding to the subject of this paper, I wish to advert to the following well-known theorem.

Let ABC be any triangle with its inscribed and circumscribed circles. Then if we take any other point D in the exterior circle, and draw the tangents DE, EF, FD, the last tangent will come again to the original point D. A similar theorem is true of a polygon of any number of sides which is both inscribed in, and circumscribed to, a circle. Also if ellipses are substituted for circles. Of these theorems I remember to have seen a demonstration founded on the theory of elliptic integrals. But this appears to me to be a great waste of analytic power; for the theorem really results from first principles, as I think will be manifest from the following considerations.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 23 , Issue 2 , 1863 , pp. 285 - 298
Copyright
Copyright © Royal Society of Edinburgh 1863

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References

page 287 note * It is unnecessary to say with Chasles that they must be polygons of the same number of sides, for they cannot be otherwise. M. Chasles, in his paper here referred to, gives no demonstrations. I am not aware whether they were subsequently published.

page 291 note * These properties, viz., that and that PN = a−b are proved by Brinkley in quite a different manner (Trans. of the Royal Irish Academy, vol. ix. p. 146, &c.). He likewise proves, that if CQ produced cuts in O, the circle described on the axis major as diameter, a perpendicular let fall from O on the axis, cuts the ellipse in Fagnani's point. But I have shown that CQ produced is the asymptote of the hyperbola, an ordinate to the ellipse at Fagnani's point, passes through the intersection of the asymptote and circle. In other words, the common chord of the ellipse and hyperbola, being produced, becomes the common chord of the circle and the two asymptotes.