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Published online by Cambridge University Press: 20 January 2009
If a circle meets the sides BC, CA, AB of a triangle (Fig. 6) in the pairs of points L and l, M and m, N and n respectively, it is obvious that the pairs of connectors Mn and mN, Nl and nL, Lm and lM are antiparallel with respect to the angles A, B, C respectively.
Mr Third's Papers were received on 29th March. Sections I. and II. of the First Paper, in a different form, were read at the November and December Meetings.
† Since writing this paper I have found that a considerable number of the theorems contained in Part I. are given in two English memoirs, which I had somehow overlooked, viz., “The Relations of the Intersections of a Circle with a Triangle,” by H. M. Taylor, Proc. of the London Math. Soc., Vol. XV., pp. 122–139, in which the treatment is mainly analytical, and “Some Geometrical Proofs of Theorems connected with the Inscription of a Triangle of constant Form in a given Triangle,” by M. Jenkins, Quarterly Journal, Vol. XXI., pp. 84–89. Among the theorems in Part I. which seem to be new, the most important, perhaps, are those referring to the connection between the S-point and the angles of a system, and to the relation of the systems to coaxal systems. The special cases of Arts. 19 and 22, Part II., are also referred to in the above-mentioned papers, but in the former of these cases the S-point is not determined.
page 73 note * This construction is given by Mr Jenkins in the paper already cited.
page 82 note * See note at end of paper.
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