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Published online by Cambridge University Press: 20 January 2009
Definition. The isogonals of the medians of a triangle are called the symmedians
If the internal medians be taken, their isogonals are called the internal symmedians or the insymmedians; if the external medians be taken, their isogonals are called the external symmedians, or the exsymmedians
* See Proceedings of Edinburgh Mathematical Society, XIII. 166–178 (1895)Google Scholar
† This name was proposed by MrD'Ocagne, Maurice as an abbreviation of la droite symétrique de la médiane in the Nouvclles Annalcs, 3rd series, II 451 (1883).Google Scholar It has replaced the previous name antiparaUel median proposed by MrLemoine, E. in the Nouvelles Annales, 2nd series, XII 364 (1873).Google Scholar MrD'Ocagne, has published a monograph on the Symmedian in MrDe Longchamps's, Journal de Mathdmatiques Élémentaires, 2nd series, IV. 173–175, 193–197 (1885)Google Scholar
‡ The names symédiane intérieure and symédiane extérieure are used by MrThiry, Clément in Le troisiéme livre de Géométrie, p. 42 (1887)Google Scholar
§ MrDe Longchamps, in his Journal de Mathématiques Élémentaires, 2nd series, V. 110 (1886).Google Scholar
* MrD'Oeagne, Maurice in Journal de Muthimntlqucs Élémentairex et Spéciales, IV. 539 (1880).Google Scholar This construction, which recalls Euclid's pons asinorum, is substantially equivalent to a more complicated one given by Const. IIarkema of St Petersburg in Schlomilch's Zeitschrift, XVI. 168 (1871)Google Scholar
* DrWetzig, Franz in Schlömileh's Zeitschrift, XII. 288 (1867)Google Scholar
† MrD'Ocagne, Maurice in the NoureUes Annulet, 3rd series, II. 464 (1883)Google Scholar
‡ See Proceedings of the Edinburgh Mathematical Society, XIII. 39 (1895)Google Scholar
§ Proceedings of the Edinburgh Mathematical Society, XIII. 172 (1895)Google Scholar
‖ Educational Times, XXXVII. 211 (1884)Google Scholar
* DrWetzig, Franz in Schlömilch's Zeitschrift, XII. 288 (1867)Google Scholar
† Adams's, C. Eigenschaften des…Dreiceks, p. 2 (1846)Google Scholar
‡ MrThiry's, Clément Le troisiéme livre de Céeomérie. p. 42 (1887)Google Scholar
* DrWetzig, in Schlomilch's Zeitschrift, XII. 288 (1867)Google Scholar
* See Proceedings of the Edinburgh Mathematical Society, I. 14 (1894)Google Scholar
† Adams's, C. Eigenschaften des…Vreiecks, p. 5 (1846)Google Scholar
‡ Ivory, in Leybourn's Mathematical Repository, new series, Vol. I. Part I. p. 26 (1804).Google Scholar Lhuilier, in his Élément d'Analyse, p. 296 (1809) proves that RW:EV=sinC :sinBGoogle Scholar
* Adams's, C. Eigemchaften des … Dreiecks, pp. 3–4 (1846).Google Scholar Pappus in his Mathematical Collection, VII. 119 gives the following theorem as a lemma for one of the propositions in Apollonius's Loci Plani: If AB2:AC2=BR′ :CR′ then BR′CR′=AR′2Google Scholar
* Grebe, E. W. in Gruuert's Archir, IX. 258 (1847)Google Scholar
* “Yanto” in Leybourn's Mathematical Repository, old series, III. 71 (1803).Google Scholar See Procecdiivjs of the Edinburgh Mathematical Society, XI. 92–102 (1893)Google Scholar
* Adams's, C. Eigenschaften des…Dreiecks, p. 3–4 (1846)Google Scholar
* ProfessorNeuberg, J. in Mathesis, I. 173 (1881)Google Scholar
* The concurrency may be established by the theory of transversals
* By ProfessorNeuberg, J.. Gergonne, J. D. (1771–1859) was editor of the Annales de Mathématiques from 1810 to 1831Google Scholar
† Many of the properties of the J points were given by Nagel, C. H. in his Untersuchungen über die wichtigsten zum Dreiecke gehöriigen Kreise (1836). This pamphlet I have never been able to procure. Since 1836 some of these properties have been rediscovered several timesGoogle Scholar
* Godwaid, William in the Lady's and Gentleman's Diary for 1807, p. 63. He contrasts this point, in reference to one of its properties, with the centroid of ABC, and recognises it as the point determined by Mr Stephen Watson in 1865. See § 8 (2) of thin paper.Google Scholar
* Godward, William in Mathematical Questions from the Educational Times II. 87, 88 (1865)Google Scholar
* Geometricus (probably MrGodward, William) in Mathematical Questions from the Educational Tines, III. 29–31 (1865). The method of proof is not his.Google Scholar
Mr W. J. Miller adds in a note that I1A′ divides I2I3 into parts which have to one another the duplicate ratio of the adjacent sides of the triangle I1I2I3 and similarly for I2B′ I3C′ ; and that the point of concurrency is such that the sum of the squares of the perpendiculars drawn therefrom on the sides of the triangle I1I2I3 is a minimum, and these perpendiculars are moreover proportional to the sides on which they fall.
† ProfessorDöttl, Johann in his Neue merkwürdige Punkte des Dreiecks, p. 14 (no date) states the concurrency, but does not specify what the points are.Google Scholar
‡ Adams's, C. Eigenschaftcn des…Dreiecks, p. 5 (1846)Google Scholar
* The first of these is mentioned by DrWetzig, Franz in Schlomilch's Zeitschrifl, XII. 289 (1867)Google Scholar
† This is one of Apollonius's theorems. See his Conies, Book III., Prop. 37–40Google Scholar
‡ Adams's, C. Eigenschaften des…Dreiecks, pp. 3–4 (1846)Google Scholar
* MrMathieu, J. J. A. in Nouvelles Annales, 2nd series, IV. 404 (1865)Google Scholar
* DrWetzig, Franz in Schlömilch's Zeittchrift, XII. 289 (1867)Google Scholar
† Grebe, E. W. in Grunert's Arehiv, IX. 253 (1847)Google Scholar
* DrWetzig, Franz in Sehlömilch's Zeitschrifl, XII. 297 (1867)Google Scholar
* MrLemoine, Emile in the Journal de Mathématiques Élémentaires, 2nd series, III. 52–3 (1884)Google Scholar
† This theorem and the proof of it have been taken from ProfessorFuhrmann's, W. Synthetische Beweise planimetriseher Sätze, pp. 101–2 (1890)Google Scholar
* For proof of some of the statements made here, see Proceedings of the Edinburgh Mathematical Society, I. 36–7 (1894)Google Scholar
† DrWetzig, Franz in Schlömilch's Zeittchrift, XII. 298 (1867)Google Scholar
* DrCasey, John in Proceedings of the Honed Irish Acadami, 2nd series, IV. 546 (1886)Google Scholar
† Rev. Simmons, T. C. in Milne's Companion to the Weekly Problem Papers, p. 150 (1888)Google Scholar
* Adams, C. in his Eiyenxchaftcn dis…Dirkcis, pp. 4–5 (1846) gives (3)—(7)Google Scholar
* MrLemoine, at the Lille meeting (1874) of the Association Fmnraise pour Vavanrchtcat dcs sciencesGoogle Scholar
† MrWatson, Stephen in the Lady's and Gentleman's Diary for 1865, p. 89, and for 1866, p. 55Google Scholar
* MrWatson, Stephen in the Lady's and Gentleman's Diary for 1866, p. 55Google Scholar
* MrMillbourn, Thomas in the Lady's and Gentleman's Diary for 1867, p. 71, and for 1868, p. 75Google Scholar
* This mode of proof is due to MrDavis, E. F.. See Fourteenth General Report (1888) of the Association for thy Improvement of Geometrical Teaching, p. 39.Google Scholar
* Properties (6)—(9) are due to Mr Tucker. See Quarterly Journal, XIX. 344, 340 (1883)Google Scholar
* DrCasey, John. See his Sequel to Euelid, 6th ed., p. 190 (1892)Google Scholar
* See the Quarterly Journal, XX. 57–59 (1884)Google Scholar
* The properties (1), (2), (3), (6), (7) are due to Mr Tucker. See Quarterly Journal, XX. 59, 57, XIX. 348, XX. 59 (1884, 3)Google Scholar
* The property that Y1Z1 is equal to the semiperimeter of XYZ occurs in Lhuilier's, Élémeas d' Analyse, p. 231 (1809)Google Scholar
† The first of these equalities is given by Feuerbach, , Eigenschaften des … Dreiecks, §19, or Section VI., Theorem 3 (1822).Google Scholar The other three are given by Hellwig, C. in Grunert's Archiv, XIX., 27 (1852).Google Scholar The proof is that of Messrs Heal, W. E. and Mange, P. F. in Artemas Martin's Mathematical Visitor, II. 42 (1883)Google Scholar
* DrKiehl, of Bromberg, . See his Zur Thcoric dcr Transversalen, pp. 7–8 (1881).Google Scholar See also Proecedings of the London Mathematical Socicty, XV. 281 (1884)Google Scholar
* This corollary and the mode of proof have been taken from DrCasey's, John Sequel to Euclid, 6th ed. p. 195 (1892)Google Scholar
† DrCasey's, John Sequel to Euclid, 5th ed. p. 195 (1892)Google Scholar
* Adams's, C. Die Lehre von den Transvcrsalen, pp. 77–80 (1843)Google Scholar
* This method of proof is different from Adams's
* Adams's, C. Die Lehre von den Transversalen, p. 79 (1843)Google Scholar
* MrThiry, Clément, Applications remarquables du TMorime de Stewart, p. 20 (1891)Google Scholar
* The values of AK BK CK are given by Grebe, E. W. in Grunert's Archiv, XI. 252 (1847)Google Scholar
† DrWetzig, Franz in Schlömilch's ZeiUchrift, XII. 293 (1867)Google Scholar
‡ MrThiry, Clément, Applications remarquatilet de Théoréme de Stewart, p. 38 (1891)Google Scholar
* DrWetzipr, Franz in Sohlömilch's Zeitsehrift, XII. 294 (1867)Google Scholar
* Grebe, E. W. in Grunert's Archiv, IX. 252, 250–1 (1847)Google Scholar
* MrTucker, R. in Quarterly Journal of Mathematics, XIX. 342 (1883)Google Scholar
† The first of these values is given by “Yanto” in Leybourn's, Mathematical Repository, old series, Vol. III . p. 71 (1803).Google Scholar Lhuilier, in his Élémens d' Analyse, p. 298 (1809) gives the analogous property for the tetrahedron.Google Scholar
The other values are given by Grebe, E. W. in Grunert's Archie, IX. 251 (1847)Google Scholar
* DrWetzig, Franz in Schlömiloh's Zeitschrift, XII. 294–295 (1867)Google Scholar
† Both forms are given by Grebe, E. W. iu Grunert's Archiv, IX. 253 (1847)Google Scholar
* DrWetzig, Franz in Schlöimilch's Zeitschrift, XII. 298 (1867)Google Scholar
† Schulz, L. C. von Strasznicki in Baumgartner and D'Ettingshausen's Zeitschrift fur Physik und Matlicmatik, II. 403 (1827)Google Scholar
‡ DrWetzig, Franz in Schlömilch's Zeitschrift, XII. 287, 293, 291 (1807)Google Scholar
* DrWetzig, in Schlöimilcli's ZciUchrift, XII. 298, 296, 292 (1867)Google Scholar
* It was this property which suggested to Mr Tucker the name “triplicateratio circle.”
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