# Exploring Special Lines (Pappus, Desargues, Pascal's Mystic Hexagram)

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```					Project AMP    Dr. Antonio R. Quesada – Director, Project AMP

Exploring Special Lines
(Pappus, Desargues, Pascal’s Mystic Hexagram)

Introduction
These three lab activities focus on some of the discoveries made by famous
mathematicians by investigating lines.
The first activity focuses on the work by Desargues. Desargues (1591-1661) was
institutions. Although recognized for his designs of a spiral staircase and other
inventions, he is best known for his work in geometry. He is considered the inventor of
projected geometry.
Pappus’ Line is the focus of the second lab. Pappus (290-350) lived in
Alexandria and was one of the famous Greek geometers. He wrote Mathematical
Collection and is considered the father of projective geometry.
The third lab allows students to investigate Pascal’s mystic hexagram.
Frenchman Blaise Pascal (1623-1662) was most famous for the Pascal triangle and his
work on the cycloid. He discovered his mystic hexagram at the age of 16 and read
Euclid’s Elements at the age of 12.

Key Words:
Lines, modern geometry

Ohio State Model Curriculum Objectives
1) Students will be able to compare, order, and determine equivalence of real
numbers.
2) Students will be able to write inequalities for various triangular relationships.

Learning Objectives
1) Students will complete basic constructions with a series of lines.
2) Students will make conjectures and test their results using geometry software.
3) Students will discuss collinear and non-collinear points.

Materials
Computers or calculators with Cabri geometry installed
Exploring Special Lines lab worksheet

Procedures
Discuss the mathematicians Pappus, Desargues, and Pascal.
Divide the students into groups of no more than three (two is preferable)
Have students complete the lab worksheet.
Monitor students’ progress during the activity.
Use results from the lab activity as assessment.
Review findings prior to the conclusion of class.
Project AMP      Dr. Antonio R. Quesada – Director, Project AMP

Exploring Special Line - Desargues Line
Lab #1 Worksheet

Team members: _______________________________________________________
File Name: ___________________ Date: ____________________

Lab Goals
Students will investigate a famous result discovered by Desargues through his
work with lines. Desargues (1591-1661) was from a wealthy family and was well-
educated. Although recognized for his designs of a spiral staircase and other inventions,
he is best known for his work in geometry. He is considered the inventor of projected
geometry.

Procedures
1) Draw point A.                                           (use point tool)
2) Construct three different lines VBCD and VEFG . Do not put much space
between the different points.                                     (use line tool)

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3) Place points E, F, and G on lines AB, AC, and AC , respectively.
(use point on object tool)

4) Construct triangle VBCD and VEFG .                    (use triangle tool and attribute tool)

Make the triangle thick to stand
out.

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5) Construct lines BC, BD, CD, EF , FG, EG .                       (use line tool)

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6) Find the intersection of lines BC and EF . Label the point H. Next, find the
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intersection of lines BD and EG. Label this point I. Finally, find the intersection
Project AMP      Dr. Antonio R. Quesada – Director, Project AMP

suur   suur
of lines CD and FG . Label this point J.                      (use intersection points tool and
attribute tool)

7) What do you notice about the points H, I, and J? _____________________

_______________________________________________________________

_____________________________________________________________

9) Now grab a vertex of triangle V BCD and move it around. Record your
observations below. ______________________________________________

__________________________________________________________________

10) Repeat #9 using triangle VEFG . Record your observations.
______________________________________________________________

_________________________________________________________________

Triangles VBCD and VEFG are defined as being homological (or in
perspective). Since all three lines pass through point A, point A is described as
the homological center. The line that contains points H, I, and J is the
homological axis.

Extension
Draw a line connecting H, I, and J. Hide everything except point A, the two
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triangles and the homological axis (line HJ ).
Experiment by moving the vertices of the
triangles and making generalizations about the
effects of moving each vertex. Record your
thoughts. Also, describe why the triangles are
described as being in perspective. Discuss
why the terms homological axis and
homological center are used.
Project AMP      Dr. Antonio R. Quesada – Director, Project AMP

Exploring Special Lines - Pappus Line
Lab #2 Worksheet

Team members: _______________________________________________________
File Name: ___________________ Date: ____________________

Lab Goals
Students will investigate a famous result discovered by Pappus through his work
with lines. Pappus’ Line is the focus of this lab. Pappus (290-350) lived in Alexandria
and was one of the famous Greek geometers. He wrote Mathematical Collection and is
considered the founder of projective geometry. It is actually a special case of Pascal’s
Mystic Hexagram, which is the subject of the next lab.

Procedure
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1) Construct line AB . Place point C on the line.
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(use line tool and use objects on line tool)

2) Draw line DE . Place F on the line.                      (use line tool and use objects on line tool)

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3) Draw lines AD , and EC . Label the intersection of those lines point G.
(use line tool and use intersection point tool)

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4) Draw lines BE and AF . Label the intersection of those lines point H.

(use line tool and use intersection point tool)

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5) Draw lines CF and DB . Label the intersection of these lines point I.
(use line tool and use intersection tool)
Project AMP      Dr. Antonio R. Quesada – Director, Project AMP

6) Record your observations below. Focus on the points G, H, and I. What do
you notice?
_______________________________________________________________

_______________________________________________________________

7) Test your results from above. Is your conjecture from #7 correct?
__________________________

8) Grab the different points and move them around. Record any other
observations that you see. _________________________________________

_________________________________________________________________

Extension
Connect the points G, H, and I. Hide the lines and watch the points as
they move. Describe the locus of points as they move around the screen. Record
Project AMP      Dr. Antonio R. Quesada – Director, Project AMP

Exploring Special Lines – Pascal’s Mystic Hexagram
Lab #3 Worksheet

Team members: _______________________________________________________
File Name: ___________________ Date: ____________________

Lab Goals
Students will investigate a famous discovery of Pascal through his work with lines
- the mystic hexagram. Frenchman Blaise Pascal (1623-1662) was most famous for the
Pascal triangle and his work on the cycloid. He discovered his mystic hexagram at the
age of 16 and read Euclid’s Elements at the age of 12. The Pappus line in lab #2 is a
special case of Pascal’s work.

Procedure
1) Construct a circle of any size. Place six points on the circle in the following
order: A, B, C, D, E, and F.                      (use circle tool and use points on object tool)

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2) Draw lines DA and EB . Label the intersection of these lines point G.

(use line tool and use intersection point tool)

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3) Draw lines AC and BF . Label the intersection of these lines points H.

(use line tool and use intersection point tool)
Project AMP      Dr. Antonio R. Quesada – Director, Project AMP

suur   suur
4) Draw lines CE and FD . Label the intersection of these points I.

(use line tool and use intersection point tool)

5) Record your observations below. Focus on the points G, H, and I. What do
you notice?
_______________________________________________________________

_______________________________________________________________

6) Test your conjecture from #7. Is your conjecture from #7 correct?
__________________________

7) Grab the different points and move them around. Record any other
observations that you see. _________________________________________

_________________________________________________________________
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The line HG is known as the Pascal line .

Extension
Connect the points G, H, and I. Hide the lines and watch the points as
they move. Describe the locus of points as they move around the screen. Record