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DESARGUES'S TWO-TRIANGLE THEOREM
Sandra Lach Arlinghaus
Often the introduction of added
perspective from another dimension sheds light on pattern. So it
is with the following theorem from Projective Geometry which becomes
easy to visualize in three dimensions. Google Earth will assist
in creating graphics for such visualization.
THEOREM [Coxeter, 1961]: If two triangles, A, B, C, and A', B',
C' are perspective from point O (AA',
BB', and CC' are concurrent),
then the intersections of corresponding triangle sides, L=AB'.A'B,
M=AC'.A'C, and N=BC'.B'C,
- Figure 1 shows an animated
illustration of Desargues's Theorem in the plane, only.
1. Desargues's Two-Triangle Theorem visualized in the plane.
- Figure 2 shows an animated
illustration of Desargues's Theorem in the plane, with background
derived from Google Earth.
2. Desargues's Two-Triangle "Tower" centered on the Sidney Smith
Building, home of the Mathematics Department of the University of
Toronto and the academic home of Professor H. S. M. Coxeter for many
- Figure 3 is a .kmz file which must be
viewed in Google Earth. It permits the reader to fly around the
Leaning "Tower" of Desargues and examine positions of triangle
Download the kmz file here.
visualize the theorem in the plane while flying through the .kmz file,
configuration so that it appears to flatten out in the plane.
When triangle sides are "parallel" in the Euclidean sense the theorem
still holds. The lines forming the sides intersect at points at
infinity, which, in the projective plane, are no different from any
others. The proof of this theorem is often presented in three,
rather than in two, dimensions.
H. S. M. 1961. Introduction
to Geometry. New York: John Wiley & Sons.
- Google Earth.
SketchUp Pro, version 6.
F. R. "A simple non-Desarguesian plane geometry," Transactions
of the American Mathematical Society,
Vol. 3, No. 2 (Apr., 1902), pp. 192-195
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An Electronic Journal of Geography and Mathematics,
XVIII, Number 2
of Mathematical Geography (IMaGe).
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