If the three straight lines joining the corresponding vertices of two triangles 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. Coxeter,
H. S. M. and Greitzer, S. L. "Perspective Triangles; Desargues's
Theorem." §3.6 in Geometry
Revisited. Durell,
C. V. Modern
Geometry: The Straight Line and Circle. Eves,
H. "Desargues' Two-Triangle Theorem." §6.2.5 in A
Survey of Geometry, rev. ed. Graustein, W. C. Introduction
to Higher Geometry. Ogilvy,
C. S. Excursions
in Geometry. Johnson,
R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Desargues' Theorem
Cite this as:
Weisstein, Eric W. "Desargues' Theorem."
From MathWorld https://mathworld.wolfram.com/DesarguesTheorem.html
Subject classifications
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.
configuration sometimes
called Desargues' configuration.
