**Pappus' Theorem**

(Geometric Structures)

prepared by:

Joseph Malkevitch

Department of Mathematics and Computing

York College (CUNY)

Jamaica, New York 11451

Closely associated with Desargues' Theorem is the wonderful theorem of Pappus. Pappus' Theorem (projective plane version) states that given the three points A, B, and C on line l and A', B', and C' on line m that if segments AB' and A'B meet at R, BC' and B'C meet at P, and CA' and C'A meet at Q then P, Q, and R are collinear (e.g. lie on a line).

These two theorems were shown to have an unexpected connection with algebra. It turns out that one can introduce a coordinate system for a geometry where the coordinates are drawn from a field provided that Pappus' Theorem holds. One can introduce a system of coordinates where all the requirements of a field except commutativity of multiplication holds provided that Desargues' Theorem holds. An algebraic system of this kind is known as a division ring. It turns out that that due to a theorem of Wedderburn, every finite division ring is a field. Thus, in a finite projective plane where Desargues' Theorem holds, then Pappus' Theorem holds. Finite non-desarguian planes can not be coordinatized with numbers from a field.

Desargues was French and lived from 1591-1661. Pappus of Alexandria (Egypt) flourished circa 300 A.D.

1. Given the point A = (2, 6, 1/2) in the real projective plane:

a. Find other coordinates for this point which are all integers.

b. Find other coordinates for this point which are all negative rational numbers.

c. Find coordinates for A whose first coordinated is 1.

2. If B = (1, -1, 1) and C = (1, 2, 1)

a. Find the equations of the lines AB, AC, and BC.

b. Find the points where each of these lines meets the line at infinity.

3. Given the triangles:

A = (2, 3, 1), B = -1, 3, 1), C = (3, 7, 1)

and

A' = (4, 7, 1), B' = (4, 6, 5), C' = (4, 2, 5)

a. Find the equation of AA'

b. Find the equation of BB'

c. Find the equation of CC'

d. Does these three lines go through a single point? If so give its coordinates.

e. Find the point X where AB and A'B' meet.

f. Find the point Y where AC and A'C' meet.

g. Find the point Z where BC and B'C' meet.

h. Do these three points X, Y, and Z lie on a line?

3. Suppose A = (3, 7, 1), B = (4, 2, 5) and C = (5, 1, 7)

and A' = (-1, 3, 1) and B' = (4, 6, 5) and C' = (1, 7, 4)

a. Determine if A, B, and C are collinear.

b. Determine if A', B', and C' are collinear.

c. If the hypothesis of Pappus' Theorem holds, verify the conclusion.