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Desargues' Theorem and SSASS

Date: 12/15/98 at 18:47:30
From: Brian Walsh
Subject: Desargues + congruence postulate

Actually I have two questions:

1) Using constructions, determine whether SSASS (Side, Side, Angle, 
   Side, Side) is a congruence postulate for Quadrilaterals.

2) With a ruler, draw 3 rays, OP, OQ, OR. Let A be any point on OP.  
   Let B be any point on OQ. Let C be any point on OR. Draw 6 lines: 
   AB, BC, AC, PQ, PR, and QR. Verify the incredible discovery of 
   Gerard Desargues, in the early 1600's, that either:

   (a) AB is parallel to PR or BC is parallel to QR or AC is parallel 
       to PQ.  

   Or (b) the 3 points of intersection are collinear.  

(This result is known as the Desargues Theorem.)

Date: 12/16/98 at 03:29:47
From: Doctor Floor
Subject: Re: Desargues + congruence postulate

Hi Brian,

Thanks for your question.

For your first question consider the following two figures:

              *               *
           * *             *   *
        *   *           *       *
     A     *         A           *
        *   *           *       *
           * *             *   *
              *               *

Although angle A has the same measure in both of these quadrilaterals, 
you can see that they have equal SSASS, but are not congruent. I think 
you will be able to show this using constructions!

For Desargues Theorem consider PQR.O as (the projection of) a pyramid. 
In this pyramid PQR and ABC form two planes. If the two planes are 
parallel, then you have all three parallelisms of (a). If the two 
planes intersect, then the intersection line can be parallel to one of 
the sides of PQR. Then you get one of the parallelisms of (a). In other 
cases AB and PQ meet at the line of intersection, and so do BC and QR, 
and AC and PR. And we have case (b).

If you have a math question again, please send it to Dr. Math.

Best regards,

- Doctor Floor, The Math Forum   
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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