Geometric Constructions

Contributed by:
This pdf includes the following topics:-
Philosophy of Constructions
Compass vs. Dividers
Basic Constructions
Constructible Numbers
Impossibility Proofs
1. Geometric Constructions
2. Philosophy of Constructions
Constructions using compass and straightedge have
a long history in Euclidean geometry. Their use
reflects the basic axioms of this system. However,
the stipulation that these be the only tools used in a
construction is artificial and only has meaning if
one views the process of construction as an
application of logic. In other words, this is not a
practical subject, if one is interested in constructing
a geometrical object there is no reason to limit
oneself as to which tools to use.
3. Philosophy of Constructions
The value of studying these constructions lies in
the rich supply of problems that can be posed in this
way. It is important that one be able to analyze a
construction to see why it works. It is not important
to gain the manual dexterity needed to carry out a
careful construction.
4. Compass vs. Dividers
The ancient Greek tool used to construct circles is
not the modern day compass. Rather, they used a
device known as a divider. Dividers consist of just
two arms with a central pivot. Should you pick up a
divider, the arms will collapse, so it is impossible to
use them to transfer lengths from one area to
another. Modern compasses remain open when
picked up, so such transfers are possible. Given the
difference in the two tools, it appears that the
modern compass is a more powerful instrument,
capable of doing more things.
5. Compass vs. Dividers
However, this is not true.
The ancient dividers can do everything that modern
compasses can. Of course, this means that how
certain constructions were done by the ancient
Greeks are quite different from the way we would
do them today. This underscores the statement
above; technique is not as important as
understanding why it works.
6. Basic Constructions
The basic constructions are:
1. Transfer a segment.
2. Bisect a line segment.
3. Construct a perpendicular to a line at a point on the line.
4. Construct a perpendicular to a line from a point not on
the line.
5. Construct an angle bisector.
6. Copy an angle.
7. Construct a parallel to a line through a given point.
8. Partition a segment into n congruent segments.
9. Divide a segment into a given ratio (internal and
7. Basic Construction 1
Transfer a line segment.
8. Basic Construction 2
9. Basic Construction 3
10. Basic Construction 4
11. Basic Construction 5
12. Basic Construction 6
13. Basic Construction 7
14. Basic Construction 8
15. Basic Construction 9
16. Basic Construction 9
17. Constructible Numbers
Given a segment which represents the number 1 (a unit segment),
the segments which can be constructed from this one by use of
compass and straightedge represent numbers called Constructible
Numbers. Note that the restrictions imply that the constructible
numbers are limited to lying in certain quadratic extensions of the
Given two constructible numbers one can with straightedge and
compass construct their:
Square Root
18. Constructible Numbers
a + b
| |
b b–a
19. Constructible Numbers
a a/b
b 1 b
20. Constructible Numbers
Square Root
a 1
21. Constructions
Example: Construct a triangle, given the length of one side of the
triangle, and the lengths of the altitude and median to that side.
22. Constructions
As the third vertex is determined by the intersection of one of two parallel
lines with a circle, there are three possibilities for the number of solutions. If b
is less than c, there will be no intersection, so no solutions. If b equals c, the
lines will be tangent to the circle and we would get two solutions. Finally, if b
is greater than c (the situation drawn above) then there will be four points of
23. Constructions
Example: Construct a triangle, given one angle, the length of the side opposite
this angle, and the length of the altitude to that side.
24. Constructions
As the position of vertex A is determined by the intersection of a single line
with a circle, there are three possibilities for the number of solutions. If the
parallel does not intersect the circle, there is no solution. If the parallel is
tangent to the circle there is one solution, and finally, if the parallel intersects
the circle twice, there are two solutions (as indicated in the situation drawn
25. Constructions
Example: Construct a triangle, given the circumcenter O, the center of the
nine-point circle N, and the midpoint of one side A'.
26. Constructions
This construction always gives a unique triangle provided one exists. If N = A'
there will be no nine-point circle, but N could equal O, or A' could equal O
and the construction will still work. The points could also be collinear.
27. Impossibility Proofs
An algebraic analysis of the fields of constructible
numbers shows the following:
Theorem: If a constructible number is a root of a cubic
equation with rational coefficients, then the equation must
have at least one rational root.
While we will not prove this result, we shall use it to
investigate some old geometric problems that dealt with
28. Impossibility Proofs
The three famous problems of antiquity are:
The Delian problem - duplicating the cube. The problem
is to construct a cube that has twice the volume of a given
cube. A particular instance of this problem would be to
construct a cube whose volume is twice that of the unit
cube. This entails constructing a side of the larger cube,
and in this case that means constructing a length equal to
the cube root of 2. This length is a root of the equation
x - 2 = 0, but this cubic equation with rational coefficients
has no rational root.
29. Impossibility Proofs
Trisection of an Angle - The problem is to find the angle
trisectors for an arbitrary angle. The general problem can
not be done because it can't be done for some specific
angles, for instance an angle of 60º. (Construction of a 20
degree angle leads to the cubic equation 8x -6x - 1 = 0,
and this does not have roots of the required type).
30. Impossibility Proofs
Squaring the Circle - The problem is to construct a square
that has the same area as the unit circle, i.e. π. If this can
be done, then the square root of π would be constructible.
And if that is true, then π would also be constructible. But
π is a transcendental number (Lindemann, 1882), and
such numbers are not constructible.
31. Angle Trisection
Angle Trisection can be done in many ways, some of
which were known to the ancient Greeks. A simple
method which uses a marked straightedge is due to
Archimedes (287-212 B.C.) and another uses the
Conchoid of Nichomedes (240 B.C.).
32. Archimedes' Angle Trisection
33. Archimedes' Angle Trisection
Let ∠AOT = x. ∠AOT ≅ ∠OTB (alternate interior angles of || lines.)
∠OTB ≅ ∠TBS since ∆SBT is isosceles. ∠BSO = 2x since it is an
exterior angle which is equal to the sum of the two opposite interior
angles. ∠BOS ≅ ∠BSO since ∆BSO is isosceles. Therefore, ∠AOT is
1/3 of ∠AOB.
34. Conchoid of Nicomedes
Given a point O, a line l not through O and a length k we
form the conchoid by adding the length k to all line
segments drawn from O to l.
35. Conchoid of Nicomedes
A k/2
36. Circle Squarers
“We have not placed in the above chronology of π any items from the
vast literature supplied by sufferers of morbus cyclometricus, the circle-
squaring disease. These contributions, often amusing and at times almost
unbelievable, would require a publication all to themselves.”
- Howard Eves, An Introduction to the History of Mathematics
Circle squarers, angle trisectors, and cube duplicators are members
of a curious social phenomenon that has plagued mathematicians
since the earliest days of the science. They are generally older
gentlemen who are mathematical amateurs (although some have had
mathematical training) that upon hearing that something is
impossible are driven by some inner compulsion to prove the
authorities wrong.
37. Circle Squarers
In 1872, Augustus De Morgan's (1806-1871) widow edited and had
published some notes that De Morgan had been preparing for a
book, called A Budget of Paradoxes. A logician and teacher, De
Morgan had been the first chair in mathematics of London
University (from 1828). Besides his mathematical work, he wrote
many reviews and expository articles and much on teaching
mathematics. In the Budget, he examines his personal library and
satirically barbs all the examples of weird and crackpot theories that
he finds there. As he points out, these are just books that randomly
came into his possession – he did not seek out any of this type of
material. In the approximately 150 works he examined, there can be
found 24 circle squarers and an additional 19 bogus values of π.
38. Angle Trisectors
DeMorgan's book was very successful. Today, with a couple of
notable exceptions, there are hardly any circle squarers left.
However, their cousins, the Angle Trisectors are still with us.
Underwood Dudley, in 1987, wrote A Budget of Trisections in an
attempt to do for Angle Trisectors what DeMorgan had done for
Circle Squarers.
The comments and quotes that follow are all from Dudley's book.
(The 2nd edition came out in 1996 and was renamed The
39. Angle Trisectors
There are several characteristics of angle trisectors (shared by others
of their ilk) that may help you identify them.
1. They are men. Almost universally. Women seem to have more
2. They are old. Often retired, having led a successful life in their
chosen endeavors. Too much free time.
3. They fail to understand what “impossible” means in
mathematics. The meaning is unfortunately not the same as the
meaning in English. It is one of the great failures of mathematics
education that this essential difference is not made plain to
40. Angle Trisectors
Typical is the trisector who wrote
“I received through the mail an advertising brochure, from a
science magazine, that had in it a simple statement – and it went
something like this – the FORMULA for TRISECTING AN
ANGLE had never been worked out. This really intrigued me. I
couldn't believe, after hundreds of years of math, that this could
be true.”
So he went to the library and found that all the books agreed that
it was impossible.
“How could men of science be so stupid? Any scientist or
mathematician who declares that a thing is impossible is showing
his limitations before he even starts on the problem at hand.”
41. Angle Trisectors
Another trisector wrote in 1933:
“Moreover, we find our modern authorities of mathematics not
attempting to solve these unsolved problems, but writing
treatises showing the impossibility of proving them. Instead of
offering inducements to the solution of these problems, they
discourage others and dub them as 'cranks'. “
4. They do not know much mathematics. Often, high school is
the last place they have seen any formal mathematics.
“It was necessary to get outside of the problem to solve it, and
it was not solved by a study of geometry and trigonometry, as
the author has never made a study of these branches of learning.”
42. Angle Trisectors
5. They think the problem is important. Since Archimedes work,
there has not been any need for such a construction, yet they
persist in thinking that mathematics has been stymied by this
“It having been hitherto deemed impossible to geometrically
trisect or divide any angle into any number of equal parts, or
fractions of parts, the author of the present work has devoted
careful study to the solving of the problem so useful and necessary
to every branch of science and art, that requires the use of
“The study of technical magazines and data shows that a
solution is being sought whereby a standard construction permits
the thrice division of any given angle ...”
43. Angle Trisectors
6. They believe that they will be richly rewarded for their work. No
one has ever put up a prize for a solution.
“When the time came for me to submit this project to a
publisher, I was very much concerned about the copyright. I was
fearful that if I submitted to a publisher, they might steal the entire
trisection and I would have to go to court and try to establish my
right to the trisection.”
44. Angle Trisectors
7. They are not logical. For instance,
“Those who are skeptical should offer something more than
rhetoric or argument in order to disprove geometrical facts.
Assuming the angle and its trisectors given, the enveloping
quadrantal arc constructed, and its points of trisection found, if it be
denied that the trisectors pass through these points of equal division
on the quadrantal arc, let them show by the ruler and compasses
where these lines and points are with respect to each other on the
quadrant. If the lines constituting the respective pairs of trisectors
of both sectors do not intersect on the quadrantal arc they should
show by the ruler and compasses where they do intersect. ”
To prove him wrong you have to trisect an angle with ruler and
compass. !!!!
45. Angle Trisectors
8. They are loners. They work by themselves, sometimes using
books, but never discuss their work until it is completed. Even
though they do not communicate with each other, they do tend to
For instance, in 1754, Jean Étienne Montucla, an early French
historian of mathematics, wrote a legitimate history of the
quadrature problem. A year later, the French Academy of
Sciences was forced to publicly announce that it would no longer
examine any solutions of the quadrature problem.
46. Angle Trisectors
9. They are prolific writers. Here is what De Morgan says about
Milan whose method gave π = 3.2 in 1855:
[The circle-squarer] is active and able, with nothing wrong with him
except his paradoxes. In the second tract named he has given the
testimonials of crowned heads and ministers, etc. as follows. Louis
Napoleon gives thanks. The minister at Turin refers it to the Academy of
Sciences and hopes so much labor will be judged worthy of esteem. The
Vice-Chancellor of Oxford – a blunt Englishman – begs to say that the
University has never proposed the problem, as some affirm. The Prince
Regent of Baden has received the work with lively interest. The
Academy of Vienna is not in a position to enter into the question. The
Academy of Turin offers the most distinct thanks. The Academy della
Crusca attends only to literature, but gives thanks. The Queen of Spain
has received the work with the highest appreciation. The University of
Salamanca gives infinite thanks, and feels true satisfaction in having the
47. Angle Trisectors
Lord Palmerston gives thanks. The Viceroy of Egypt, not yet being
up in Italian, will spend his first moments of leisure in studying the
book, when it shall have been translated into French: in the mean
time he congratulates the author upon his victory over a problem so
long held insoluble. All this is seriously published as a rate in aid of
demonstration. If those royal compliments cannot make the
circumference about 2 per cent larger than geometry will have it –
which is all that is wanted – no wonder that thrones are shaky.
- Budget, Vol. 2, pp. 61-2.
48. Angle Trisectors
Now, will you know a Angle Trisector when you see
one coming? And will you know what to do?
Hint: What you do involves your legs.
No, you do not kick him!
49. Regular Polygons (Gauss)
These are only possible when the number of sides, n, is of
the form
n = 2
where the pi are distinct Fermat primes, i.e. prime numbers
of the form 2
p i1 := 2 1
The first few Fermat primes are: p 1= 3, p2= 5, p3 = 17.
Thus, it is possible to construct regular polygons of n sides
2 3
when n is: 3, 4 = 2 , 5, 6 = 2(3), 8 = 2 , 10 = 2(5), 12 =
2 4
2 (3), 15 = 3(5), 16 = 2 and 17.
50. Regular Polygons (Gauss)
The factor of a power of 2 comes from the fact that given
any regular n-gon, you can always construct a regular 2n-
gon. This is done by inscribing the n-gon in a circle and
then constructing the perpendicular bisectors of each of the
sides. Extend these to the circle and these points together
with the original vertices of the n-gon, form the vertices of
a regular 2n-gon. Repeating this will give the higher
powers of 2.
It is not possible to construct, with straightedge and
compass alone, regular polygons of sides n = 7, 9, 11, 13,
14, 18, 19, ....
51. Other types of Constructions
It can be shown that any construction that can be made
with straightedge and compass can be made with compass
alone (Mascheroni, 1797 [Mohr, 1672]). Of course one
must understand that a straight line is given as soon as
two points on it are determined, since one can't actually
draw a straight line with only a compass.
It can also be shown that any construction that can be
made with straightedge and compass can be made with
straightedge alone, as long as there is a single circle with
its center given (Steiner, 1833 [Poncelet, 1822]).
52. Paper Folding
53. Paper Folding
54. Paper Folding
Other constructions are possible, including trisecting angles.
Here is a folding of the regular nonagon (9 sided regular polygon) which is
impossible to do with straightedge and compass (from T.S. Row, Geometric
Exercises in Paper Folding).
Unfortunately, this is
only approximate, this
construction can not be
done exactly with paper