SOLSTICE: An Electronic Journal of Geography and Mathematics. (Major articles are refereed; full electronic archives available) 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.

DESARGUES'S 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, are collinear.
• Figure 1 shows an animated illustration of Desargues's Theorem in the plane, only. Figure 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. Figure 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 years.

• 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 vertices.  Download the kmz file here

To visualize the theorem in the plane while flying through the .kmz file, tilt the 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.

References:
• Coxeter, H. S. M.  1961.  Introduction to Geometry.  New York:  John Wiley & Sons.  252-254.
• Google SketchUp Pro, version 6.
• Moulton, 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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