Desargues' Theorem -- from Wolfram MathWorld
- ️Weisstein, Eric W.
If the three straight lines joining the corresponding vertices of two triangles 
and 
all meet in a point (the perspector),
then the three intersections of pairs of corresponding sides lie on a straight line (the perspectrix). Equivalently,
if two triangles are perspective
from a point, they are perspective
from a line.
The 10 lines and 10 3-line intersections form a 
configuration sometimes
called Desargues' configuration.
Desargues' theorem is self-dual.
See also
Desargues' Configuration, Duality Principle, Pappus's Hexagon Theorem, Pascal Lines, Pascal's Theorem, Perspector, Perspective Triangles, Perspectrix, Self-Dual
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References
Coxeter, H. S. M. The Beauty of Geometry: Twelve Essays. New York: Dover, p. 244, 1999.Coxeter, H. S. M. and Greitzer, S. L. "Perspective Triangles; Desargues's Theorem." §3.6 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 70-72, 1967.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 44, 1928.Eves, H. "Desargues' Two-Triangle Theorem." §6.2.5 in A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, pp. 249-251, 1965.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 23-25, 1930.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 89-92, 1990.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 231, 1929.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 77, 1986.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 54-55, 1991.
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Cite this as:
Weisstein, Eric W. "Desargues' Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DesarguesTheorem.html