Desargues Theorem and Terminology:

Two triangles ABC and DEF are perspective from a point the lines AD, BE and CF are concurrent at a point P.

a)       Two triangles with sides a, b, c and d, e, f are perspective from a line if the intersections ad, be, cf are collinear on a line p.

Desargues Theorem:  If two triangles are perspective from a point, then they are perspective from a line.

Write the converse of Desargues Theorem:                                                                                                                  

A Desargues Configuration is a figure consisting of two triangles ABC and DEF and a point P so that the triangles are perspective at P, and also the 3 points of intersection of corresponding sides of the triangles and the line p from which the triangles are perspective.  So there is a total of 10 points and 10 lines in the figure.

An experiment demonstrating the 3-dimensional proof of Desargues theorem

On the file folder, draw a line down the fold.  Then on one half of the folder draw a triangle ABC and on the other half draw a triangle DEF so that the triangles are perspective from a line.

With your compass, punch a hole at each of the six triangle vertices.

Now fold the folder and thread skewers through the points A and D, B and E, C and F.  The skewers should be approximately concurrent at a point P in space.

Desargues Questions and Problems

a)       Why is it true that the triangles in space are perspective from a point if they are perspective from a line?  Hint:  How do planes meet in space?

b)       Was this file folder construction a demonstration of Desargues or its converse?

c)       How could you argue from the same 3D figure (but with different givens) to prove the converse of the theorem you proved in (a)?

d)       Draw a figure illustrating a Desargues configuration when two triangles are perspective from the line at infinity.

e)        Draw a figure illustrating a Desargues configuration when two triangles are perspective from a point at infinity.

f)        Draw a figure illustrating a Desargues configuration when two triangles ABC and DEF are perspective from a point and when points A and D are on the line at infinity.