Pappus of Alexandria described Pappus' theorem, which states that if points A, B, and C lie on one line and corresponding points A', B', and C' lie on another line, the lines connecting these points (AB' , BA' , AC' , CA' , BC' , CB') will intersect at points that lie on a common line called the Pappus line. Girard Desargues later established projective geometry and described three theorems: 1) Two triangles are perspective from a point if lines connecting corresponding vertices meet at a common point, 2) Triangles are perspective from a line if intersections of corresponding lines all lie on a common line, and 3) Triangles are perspective

3. Pappus of Alexandria (340 A.D.)

4. Pappus’ Theorem
If points A,B and C are on one line and
A', B' and C' are on another line then
the points of intersection of the lines
AB' and BA', AC' and CA', and BC' and
CB' lie on a common line called the
Pappus line of the configuration.

5. Pappus’
Theorem
C
B
A
A'
If points A,B and C are on one line and
A', B' and C' are on another line then
the points of intersection of the lines
AB' and BA', AC' and CA', and BC' and
CB' lie on a common line called the
Pappus line of the configuration.
B'
C'

6. The Pappus’ geometry
configuration has 9 points and
9 lines.

7. Girard Desargues (1591 - 1661)
Father
of
Projective
Geometry

8. Desargues’ Theorem 1
Two triangles said to be perspective
from a point if three lines joining
vertices of the triangles meet at a
corresponding common point called
the center or polar point.

9. Desargues’
Theorem 1
Two triangles said to be perspective from a point if three
lines joining vertices of the triangles meet at a
corresponding common point called the center or polar
point.

10. Desargues’ Theorem 2
Two triangles are said to be
perspective from a line if the three
points of intersection of
corresponding lines all lie on a
common line, called the axis.

11. Desargues’
Theorem 2
Two triangles are said to be perspective from a line if the
three points of intersection of corresponding lines all lie
on a common line, called the axis.

12. Desargues’ Theorem 3
Desargues' theorem states that two
triangles are perspective from a
point if and only if they are
perspective from a line.

13. Desargues’
Theorem 3
Desargues' theorem states that two triangles are
perspective from a point if and only if they are
perspective from a line.

14. The Desargues’ geometry
configuration has 10 points and
10 lines.

15. Prepared and presented by:
AMORY RICAFORT BORINGOT