Unit 3 PG# Introduction

3.1.  Introduction

The French mathematician Gerard Desargues (1593–1662) was one of the earliest contributors to the study of Projective Geometry. A theorem known as “Desargues theorem” bears his name and often set him as inventor of projective geometry. This theorem is very useful in the development of projective geometry. It is about relating two aspects of projective geometry: pair of three vertices and pair of three sides. We deal this theorem with a couple of terms as below.

Couple

Two triangles Δabc and Δa'b'c' are said to be a couple if

                Points a,b,c,a',b',c' are distinct

                Lines aa', bb',cc' are distinct

                Lines aa', bb',cc' does not lie on either sides of triangles.

Central Couple

Two triangles Δabc and Δa'b'c' are said to be a central couple if

                Δabc  and Δa'b'c' are couple

                lines  aa',bb',cc' are concurrent
NOTE

• The point of concurrency is called center of the couple
• Two triangles which are perspective from a point is called central couple.

Axial Couple

Two triangles Δabc and Δa'b'c' are said to be an axial couple if

·         Δabc and Δa'b'c' are couple

·         points ab∩a'b', bc∩c'c', ca∩c'a' are collinear.
       NOTE

• The line of collinearity is called axis of the couple.
• Two triangles which are perspective from a line is called axial couple

3.2.  Desargues triangle theorem

If two triangles are so situated that line joining the pair of corresponding vertices are concurrent then the points of intersection of pair of corresponding sides are collinear and conversely. This theorem can be stated as “if two triangles are central couple then they are axial couple and conversely”. The Desargues theorem can also be stated as “if two triangles are perspective from a point then they are perspective from a line and conversely”. In the figure below, Δabc and Δa'b'c' satisfy Desargues triangle theorem.

Desarguesian Plane

A projective plane in which Desarguesian triangle theorem holds is called Desarguesian plane.