The Oxford, Cambridge, and Dublin Messenger of Mathematics Volume 5

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1871 edition. Excerpt: ...which touches the six lines. The transformation (as to its tangents) of a cardioidal curve into a conic and a point is illustrated by fig. 41, which represents such a curve very nearly consisting (as to its visible points) of a conic and a doubled finite portion of one of its tangents. So far we have considered the following series of propositions: (1). Given three lines, a circle may be drawn through their intersections (Euc. iv. 1). (1'). Given four lines, the four circles so determined meet in a point. (Well known). (2). Given five lines, the five points so found lie on acircle. (Miquel). (2'). Given six lines, the six circles so determined meet in a point. (Sect. 111., Cor. 2). I shall now shew that the series is interminable; that is, that 2n lines determine 2n circles all meeting in a point, and that for 2n+1 lines the 2n+ 1 points so found lie on the same circle. Connected with Prop. 1, however, there is another theorem which is susceptible of generalization. If from any point in the circumscribing circle we draw perpendiculars to the sides of a triangle, the feet of these perpendiculars are in one straight line. I call this (lp); the corresponding pendant to lliquel's theorem is, (211) If from any point p on the circle in Prop. (2) we draw perpendiculars on the five lines, their feet lie on a conic passing through p. So again we have (3). Given seven lines, the seven points obtained as in (2'), are all in one circle. (319). If from any point p on this circle we draw perpendiculars on the seven lines, their feet will lie on a cubic having a node at 12. And generally (np). If from any point p on the circle determined by 2n+ 1 lines we draw perpendiculars to them, the feet of the perpendiculars will lie on a curve of order n...show more

Product details

• Paperback | 52 pages
• 189 x 246 x 3mm | 109g
• United States
• English
• black & white illustrations
• 1236847342
• 9781236847348