Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-25T03:16:19.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Orthologic Desargues' figure

Part of: Linear incidence geometry Geometry

Published online by Cambridge University Press:  09 April 2009

Sahib Ram Mandan
Sahib Ram Mandan
Affiliation:
Flat 19, “Vitaya”, Chhedanagar, Bombay-400089, India
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1894 Sondat published a theorem that the centre of perspectivity and the 2 orthologic centres of any 2 bilogic (perspective as well as orthologic) triangles lie on a line perpendicular to their axis of perspectivity. Thébault (1952) gave an elementary proof of this theorem. Here we give two new proofs, one synthetic and the other analytic, and then deduce the existence of an orthologic Desargues' figure where all the 10 pairs of perspective triangles in it are orthologic. Consequently we arrive at an orthologic Veronese configuration of 15 points and 10 pairs of perpendicular lines studied in 5 different ways.

MSC classification

Secondary: 51A20: Configuration theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 3 , November 1979 , pp. 295 - 302
Copyright
Copyright © Australian Mathematical Society 1979

References

Baker, H. F. (1929), Principles of geometry I (Cambridge University Press).Google Scholar
Baker, H. F. (1930), Principles of geometry II (Cambridge University Press).Google Scholar
Baker, H. F. (1934), Principles of geometry III (Cambridge University Press).Google Scholar
Court, N. A. (1954), ‘Desargues and his strange theorem’, Scripta Math. 20, 1–20.Google Scholar
Court, N. A. (1964), Modern pure solid geometry (Chelsea).Google Scholar
Coxeter, H. S. M. (1964), Projectile geometry (Blaisdell, New York).Google Scholar
Coxeter, H. S. M. (1969), Introduction to geometry (Wiley, New York).Google Scholar
Coxeter, H. S. M. (1975), ‘Desargues' configurations and their collineation groups’, Math. Proc. Cambridge Philos. Soc. 78, 227–246.CrossRefGoogle Scholar
Gerber, L. (1975), ‘Associated and perspective simplexes’, Trans. Amer. Math. Soc. 201, 43–55.CrossRefGoogle Scholar
Gerber, L. (1977), ‘Associated and skew-orthologic simplexes’, Trans. Amer. Math. Soc. 231, 47–63.CrossRefGoogle Scholar
Mandan, S. R. (1960), ‘A configuration of 600 lines in [4]’, Rev. Fac. Sci. Univ. Istanbul A25, 17–44.Google Scholar
Mandan, S. R. (1965), ‘Concurrent lines related to 2 plane triangles’, J. Sci. Engrg. Res. 9, 47–50.Google Scholar
Mandan, S. R. (1966), ‘Perspective simplexes’, J. Austral. Math. Soc. 6, 11–17.CrossRefGoogle Scholar
Mandan, S. R. (1977a), ‘Skew orthologic perspective simplexes’, J. Mathematical and Physical Sci. 11, 569–576.Google Scholar
Mandan, S. R. (1977b), ‘On a Gerber's conjecture’, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (Ser. VIII) 61, 411–419.Google Scholar
Mandan, S. R. (1979), ‘Special pairs of semi-bilogic and bilogic tetrahedra’, J. Austral. Math. Soc. (Series A) 28, 303–308.CrossRefGoogle Scholar
Pedoe, D. (1963), Introduction to projective geometry (Pergamon, Oxford).Google Scholar
Room, T. G. (1967), A background to geometry (Cambridge University Press).Google Scholar
Salmon, G. (1964), Conic Sections (Chelsea).Google Scholar
Sondat, P. (1894), ‘Question 38’, L'intermédiate des mathématiciens 1,10Google Scholar
(solved by Sollerstinsky, , L'intermediaire des mathematiciens 1, 44–45).Google Scholar
Thébault, V. (1952), ‘Perspective and orthologic triangles and tetrahedrons’, Amer. Math. Monthly 59, 24–28.CrossRefGoogle Scholar
Thébault, V. (1955), Parmi les belles figures de la geometrie dans l'espace (Libraire Vuibert, Paris).Google Scholar
Veblen, O. and Young, J. W. (1938), Projective geometry I (Ginn, New York).Google Scholar