Trisections, division of angles, and construction
of regular n-gons
on Sketchpad
James M. Parks
SUNY Potsdam
There
always seems to be some confusion whenever the topic of trisection of angles
comes up in the classroom. I think
part of the problem is that many students are told "the trisection of
angles is impossible" but don't understand why, or they forget, yet at the
same time they know ways to trisect some angles, so what's the problem? The problem, of course, is in the
hypothesis (here unstated). A
correct statement would be along the lines of "it is impossible to trisect
an arbitrary angle with a compass
and unmarked straightedge", or equivalently, "it is impossible to trisect an arbitrary angle in
Euclid's geometry".
There are, however, methods which will trisect an
arbitrary angle if, say a marked
straightedge is allowed (see the method of Archimedes below). Dudley [3] gives many of these
methods and he includes a long list of examples of false trisections, that is trisections
which do not work in Euclidean geometry.
In this note we give a simple method for trisection
that can be done on Sketchpad [4]
and which leads to a general method for division of angles by odd numbers of
the form 2n+1, n a positive integer. By the way, this result points out how Sketchpad can also be used to make constructions which are
outside of Euclidean geometry.
One
of the earliest known methods for trisection is due to Hippias (485-415 BC) in
which he used the curve we call the "Quadratrix of Hippias" (see [1]
pp.3-5, [2] pp.266-7, and [3] pp.6-7).
It is recorded that Nicomedes (280-210 BC) was also aware of this curve
and used it to square the circle ([2] p.266). Nicomedes is credited with the discovery of another curve,
now called the "Conchoid of Nicomedes", which can also be used to
trisect an arbitrary acute angle ([2] pp.160-1, [3] pp.8-9). There is also recorded a method which
uses the spiral of Archimedes ([2] p.267, [3] pp.9-11). Other interesting techniques are known
and many can be found in [3].
The
trisection method of Archimedes (287-217 BC) makes use of a marked straight edge and is perhaps the simplest and
cleverest of all ([1] pp.1-3, [3] pp.4-5). Since the Sketchpad method we give below is a variation on this method we give all the
details for Archimedes' method here.
Given an acute angle <ABC,
construct a line m through a point D on AB parallel to the other leg BC, and
construct a circle with center D and radius |DB|. Mark the straight edge with the length |DB| and place the
straight edge so that it is on B and the marked points fall on the circle at F
and the line m at E. Then 3m<EBC = m<ABC. To see this connect D
to F, then triangle DBF and triangle DEF are isosceles, by construction. Also, m<DEB = m<EBC, for BC//m, and m<DBF = m<DFB = 2m<DEB = 2m<EBC, by the
Exterior Angle Theorem (Euclid's Prop. 32, [2] p.316). So EB divides <ABC in the ratio of 2:1 and we have trisected <ABC.
The following Sketchpad technique for trisection is a simple variation on the
above method of Archimedes which replaces the marked straight edge with circles
and isosceles triangles.
Let
<ABC be a given an acute angle situated as in the
sketch below (actually any angle less than 
will work).
Choose a point D on AB and construct the parallel line
m to BC through D. Choose a point
E on line m to the 'right' of D and connect it to B, so that BE divides <ABC.
Construct a circle with center D and radius |DB|, and
at the intersection F of this circle with BE construct another circle with
center F and radius |DF|.
Now move
point E to the intersection G of the 2nd circle with line m so that we have the
order B-F-E/G. The dynamic nature of Sketchpad will preserve the circle constructions. Connect D to F and observe (see sketch
below) that m<DEB = m<EBC, since BE
falls across parallel lines. Also,
m<DBF = m<DFB = 2m<DEB = 2m<EBC, by the Exterior Angle Theorem, since triangle DBF
and triangle DFE are isosceles.
The ability to 'move' point E and to take advantage of
the dynamic property of Sketchpad in the construction is what makes the method work.
To demonstrate the generalization to division of
angles by numbers of the form 2n+1, n a positive integer, we divide <ABC by 5 (n = 2).
Let <ABC be an acute angle which
opens to the 'right' with BC horizontal (see sketch below). We need to construct n+1 = 3 circles.
Choose a point D on AB and construct a parallel line m
to BC through D. Choose a point E
on this line m to the 'right' and connect E to B so that BE divides <ABC. Move E off to the
right as shown (the position of E can be adjusted as needed).
Construct a circle with center D and radius |DB|, and
at the intersection F of this circle with BE construct a 2nd circle with center
F and radius |DF|. At the
intersection H of this circle with BE, where H satisfies the order B-F-H-E,
construct a 3rd circle with center H and radius |DH|.
Move E to
the intersection J of this last circle with the line. Then we know that m<EBC = m<DEB, since m//BC, and m<DHB = 2m<DEB = 2m<EBC by the
Exterior Angle Theorem, since triangle DHE is isosceles. Similarly, since triangle DFH and triangle
DBF are isosceles, m<DBF = m<DFB = 2m<DHB = 4m<DEB = 4m<EBC, and we are
done.
The technique clearly works for any positive integer
n, for all we need to do is to keep adding circles on line BE.
This
result also gives us the means to divide angles by products of numbers of the
form 2n+1, n a positive integer. For example, if we divide <ABC by 3
to get <EBC, then divide <EBC by 5, the
result is to divide <ABC by 3x5 = 15.
Of course we could also include numbers of the form 2m by
using bisection.
Methods
for division of angles have immediate application in the construction of
regular n-gons (all sides the same length and all vertex angles the same
measure). In honor of Gauss
(1777-1855), who was the first to construct a 17-gon (at the age of eighteen)
we give a Sketchpad construction
of a 17-gon.
First let <ABC be a right
angle. Construct a circle centered
at B with radius |AB|. Let ÐEBC be the result of the division of <ABC by 17 using the technique above. Construct a line segment BP by rotation such that <PBC satisfies m<PBC = 4m<EBC. The line segments for the 17-gon are
then formed by connecting the intersection points of the circle with the rays
of <PBC and iterating this segment 16 times around the
circle using rotations by fixed angle equal to <PBC.
To see the 17-gon more clearly, hide the circle and
the division line segments (see the sketch below).
The
construction of n-gons for products of numbers of the form 2n+1,
as noted above, can also be carried out.
References
1.
Hesse, B., "Angle Trisection", Geometry Forum Articles, http://www.geom.umn.edu/docs/forum/angtri/
2.
Heath, T., "The Thirteen Books of Euclid's Elements", Vol.1, 2nd ed.,
Dover, 1956.
3.
Dudley, U., "The Trisectors", MAA Spectrum Series , 1994.
4. The
Geometer's Sketchpad, v.4.01, Key
Curriculum Press.