Download 1 Eves`s 25 Point Affine Geometry

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Eves’s 25 Point Affine Geometry
In this problem you will investigate a model for geometry which shares many properties with Euclidean
geometry, but only contains 25 points. This model is described in detail in section 1.5 of your book,
and you may find this section useful to read while completing this project.
For this problem, you must complete the following:
• As preparation for the study of Eves’s geometry, write a short description of what the following terms mean in Euclidean geometry: distance, endpoints of a segment, midpoint of a section,
perpendicular bisector, median of a triangle, altitude of a triangle, parallel lines, rectangle, parallelogram. As you are doing this you might note nice properties that this objects have in Euclidean
geometry, but you do not need to prove anything.
• Now read the definition of Eves’s geometry in section 1.5 and determine what the equivalent of
each of these terms in Eves’s geometry would be. There is nothing to prove here, but you should
explain why your terms are not arbitrary. For example, when defining parallel lines in Eves’s
geometry, you should try to find a way to describe parallel lines which would be equally true in
Euclidean geometry or Eves’s geometry (even if it isn’t a formal definition).
• Now, using the definitions you have created for Eves’s geometry, prove that the points on a
perpendicular bisector are equidistant from the endpoints of the segment it bisects. Then note
that the converse of this is not true (that is, there will be points equidistant from the endpoints
which are not on the perpendicular bisector). If you cannot prove these things, then you have
chosen the wrong definitions for these words.
• Write a short discussion of your thoughts on why Eves’s geometry might share several properties
with Euclidean geometry, even though it is constructed in a very different manner. In particular,
you should comment on whether the similarities are coincidental or whether there might be a
deeper reason for them.
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Geometry and the Continuum
In this problem you will investigate models of geometry which are not continuous. Section 1.9 of your
book talks about a simple example, you will deal with more complicated examples as well.
For this problem, you must complete the following:
• Suppose that we consider a geometry (which we call the integer geometry) where points are points
in the xy-plane of the form (p, q), where p and q are both integers, and a line is a set of these points
satisfying an equation y = mx + b or x = a, where a, b and m are constant real numbers. Call
the slope of the line the number m, or ∞ if the line is of the form x = a. Call lines perpendicular
if their slopes are negative reciprocals, or if one has slope 0 and the other has slope ∞. Call two
lines parallel if they do not intersect at any point. Show that there lines which are both parallel
and perpendicular to each other.
• Now consider a geometry (which we will call the rational geometry) where the points are of the
form (p, q), where p and q are rational numbers and lines, slopes, perpendicular lines and parallel
lines are defined in the same way as the previous example. Show that in this geometry, there are
lines which are parallel and perpendicular to each other.
• Add a distance to the rational geometry as follows: If (p1 , q1 ) and (p2 , q2 ) are the coordinates of
points in this geometry (so that p, q, r and s are all rational numbers) then the distance from
(p1 , q1 ) to (p2 , q2 ) will be:
p
(p1 − p2 )2 + (q1 − q2 )2
With this notion of distance, define a circle with radius r and center O to be the set of all points
of distance r from the center O, and the interior of the circle to be the set of all points of distance
less that r from the center O. With all of these definitions, show that it is possible to find a line
and a circle so that the line contains points on the interior of the circle but it does not intersect
the circle.
• Finally, write a short discussion about why it might be natural to avoid using irrational numbers
(think about how you write down numbers), but why at the same time we cannot avoid using
irrational numbers if we want to deal with real world geometry.
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Further Investigations in Taxicab Geometry
In this problem you will further investigate the Taxicab geometry introduced in section 3.2 of the book.
While we only looked at this model to justify assuming SAS congruence as an axiom (since the existence
of this model proves that SAS congruence cannot be proven from axioms 0-11), it may be interesting
to look at Taxicab geometry in more detail.
Recall that in taxicab geometry the distance from (x1 , y1 ) to (x2 , y2 ) is defined as |x1 − x2 | + |y1 − y2 |.
For this problem, you must complete the following:
• We will use (ABC) to indicate “the point B is between A and C in Euclidean geometry” and
< ABC > to indicate “the point B is between A and C in Taxicab geometry.” Prove that (ABC)
is true if and only if < ABC > is true. (HINT: Try to relate betweenness to simple calculations
on the coordinates of a line.)
• Use the previous result to prove that Axiom 5 is true in Taxicab geometry, and that segments and
rays are the same between Euclidean geometry and Taxicab geometry.
• Define Taxicab circles as the sets of points which have a constant distance from a center, and
Taxicab ellipses as the sets of points which have a constant sum of distances from two points
(called the focii). Describe these shapes in terms of Euclidean geometry and show that your
description is correct.
• Research the idea of a general metric in mathematics. List some examples of metrics for the plane
other than the Euclidean and Taxicab metrics. Explain what you think the general relationship
between changes of metric and shapes like circles and ellipses might be (but you do not have to
prove anything in this discussion).
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Single Elliptic Geometry
In this problem you will investigate what David Kay calls “Single Elliptic Geometry”, which was briefly
discussed in section 2.2. We can construct this geometry as follows. We will define points for single
elliptical geometry to be sets of two antipodal points from spherical geometry. Kay calls this “Single
Elliptic Geometry” because each pole corresponds to one point, as opposed to the “Double Elliptic
Geometry” of Spherical Geometry, where every pole corresponds to two points.
For this problem, you must complete the following:
• Suppose that A and B are points in spherical geometry, with A∗ and B ∗ as their antipodal points,
so that {A, A∗ } and {B, B ∗ } are points in Single Elliptic Geometry. Define the Single Elliptic
distance from {A, A∗ } to {B, B ∗ } as the minimum of the spherical distances AB, AB ∗ and A∗ B ∗ .
Show that this notion of distance satisfies Axiom 4 and the triangle inequality.
• Show that the statement “given two distinct points there is exactly one line which contains both”
is true in Single Elliptic geometry, even though it is false in Spherical geometry.
• Explain why the Plane separation postulate fails in Single elliptic geometry.
• Write a short discussion explaining why the previous point makes it difficult to define angles in
Single Elliptic geometry, but how we can nevertheless come up with some notion of angles in this
geometry. Speculate about what sorts of properties angles and triangles may have in common (or
distinct from) Euclidean and Spherical geometry.
• Even though the definition of Single Elliptic geometry may be more intimidating than that of
Spherical geometry, it is more commonly used in geometry classes and in visualization programs
than Spherical geometry. For example, the program Cinderella allows constructions in Euclidean,
Hyperbolic and Single Elliptic geometry, but not Spherical geometry. Write a brief discussion of
why you think this might be. What payoff might be worth the more complicated setup?
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Transformations and the SAS Postulate
In this problem you will investigate the relationship between rigid motions (such as translations and
reflections) and congruences. The book discusses this issue somewhat in section 3.4.
Some people find it unsatisfying to assume the SAS postulate as an axiom. We argued that it should
be an axiom because of the idea that if we placed two triangles displaying such a correspondence on top
of each other, they would exactly overlap and thus the remaining sides and angles would be the same.
But if we have this argument, perhaps the SAS postulate should be something which is proven?
Here will instead assume something about rigid motions of the plane, and use this to prove the SAS
postulate.
For this problem, you must complete the following:
• Define a motion to be a one-to-one onto correspondence from the plane to itself so that if A, B
and C correspond to A0 , B 0 and C 0 , then AB = A0 B 0 and m∠ABC = m∠A0 B 0 C 0 (and similarly
with the other distances and angles involving those points). Define a reflection over a line ` to be
a motion which associates every point off of ` to a different point off of `, and which associates
every point on the line to itself. Assume Axiom M, which is the statement that there is a unique
reflection over any line.
• Using Axiom 0-11 and M (as well as any results proven only from them, which would be the results
←→
from the first two chapters), prove that if P is a point not on AB and P 0 is the reflection of P
←→
←→
←→
over AB, then P and P 0 are on opposite sides of AB, that ∠ABP ∼
= ∠ABP 0 and that AB is the
perpendicular bisector of P P 0 .
• Using your results form the last part, prove that if 4ABC and 4DEF are triangles so that
AB ∼
= DE, BC ∼
= EF and ∠ABC ∼
= ∠DEF , then there is a series of at most three reflections
which map 4ABC to 4DEF . Explain why this implies that 4ABC ∼
= 4DEF .
• Write a short discussion comparing the two approaches to establishing triangle congruences. What
are the advantages to assuming the SAS postulate, and what are the advantages to assuming that
reflections exist? Which is easier to work with in proofs, and which makes more sense to your
intuition?
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Circles and Triangles
A common problem in classical geometry is to circumscribe a circle about a polygon or to inscribe a
circle in a polygon. To circumscribe a circle about a polygon means to find a circle which contains each
vertex of the polygon. To inscribe a circle in a polygon means to find a circle which is tangent to each
side of the polygon. For an arbitrary polygon it may or may not be possible to find such circles, but for
triangles it is always possible. In this problem you will figure out how to do this.
For this problem, you must complete the following:
• Consider a triangle 4ABC. Let ` be the perpendicular bisector of AB, m the perpendicular
bisector of BC and n the perpendicular bisector of AC. Assume that α = ∞ and that each set
of two lines intersects at exactly one point. Prove that all three of the lines intersect at one point
(in generally three lines might intersect at a triangle of points).
• Use your result from the previous section to show that it is possible to circumscribe a circle about
any triangle.
• Consider a triangle 4ABC. Let ` be the line containing the angle bisector of ∠BAC, m the line
containing the angle bisector of ∠ABC and n the line containing the angle bisector of ∠BCA.
Assume that α = ∞ and that each set of two lines intersects at exactly one point. Prove that all
three lines intersect at one point.
• Use your result from the previous section to show that it is possible to inscribe a circle in any
triangle.
• Write a short discussion talking about the difficulties of inscribing or circumscribing circles in or
about general triangles. Are there situations where you think that it may be possible to find such
circles, or situations where you think that it is definitely impossible to find such circles?
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An Alternative SSS Proof
There is a way to prove the SSS theorem using only the SAS theorem and basic facts about quadrilaterals.
We do this by considering what we will call a kite and a dart. A kite is a convex quadrilateral ♦ABCD,
so that AB = AD and BC = CD, and a dart is a non-convex quadrilateral ♦W XY Z so that W X =
W Z, XY = Y Z and Y lies in the interior of 4W XZ. Therefore generally both kites and darts are
quadrilaterals with two pairs of congruent adjacent sides, with the difference between a kite and a dart
being that a kite is convex but a dart is not.
Since you will be using your results to prove the SSS theorem, you should not use the SSS theorem or
any theorem derived from it in your work on this project.
For this problem, you must complete the following:
• Prove that in a kite or a dart, the angles two angles between sides which are not congruent are
congruent to each other. (For example, if in ♦ABCD we have AB ∼
= BC and CD ∼
= AD, then
∼
you must prove that ∠BAD = ∠BCD. You may have to break your proof into different cases for
kites and for darts.
• Show that if 4ABC and 4DEF are two triangles so that AB = DE, BC = EF and AC = DF ,
then it is possible to find a point X so that 4XBC ∼
= 4DEF and ♦ABCX is either a kite or a
dart.
• Use your work in the previous two sections to prove the SSS theorem.
• Write a discussion which describes the advantages and disadvantages of this proof of the SSS
theorem over alternative techniques. In particular, how does the shape of a kite and a dart lead
to more natural discussions about congruence in relation to rigid motions of the plane and of
symmetry?
• Finally, experiment to see if it would be possible to use quadrilaterals in a similar manner to prove
other congruence theorems (or even to argue for the plausibility of the SAS postulate). Is there a
general advantage to looking at quadrilaterals over triangles, or is it only more useful in the case
of the SSS theorem?
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Constructions and Polygons
In classical geometry shapes are only considered to be constructed if they can be formed using a compass
and a straightedge. This is equivalent to saying that if we have a line segment and a point, we can
draw a circle with that point as center and a radius the same length of the segment (by using a compass
widened to the width of the segment at the point to draw a circle), and we can connect two points into
a segment or extend a line segment into a line (by using a straightedge). Additionally, we can identify
points of intersection between shapes that we have drawn. But we cannot draw rays at arbitrary angles
or extend segments by arbitrary lengths, since we do not have a protractor or ruler.
To give you an idea of how this works, the classical argument for finding the perpendicular bisector of
a segment AB is as follows: First draw a circle with center at A and radius AB. Then draw a circle
with center at B and radius AB. These circles will intersect at two points P and Q. Connect P and
Q as a segment and extend the segment into a line. The line will be the perpendicular bisector of the
segment AB.
Since this is a question about classical geometry, you can work in Euclidean geometry throughout this
problem.
For this problem you will need to complete the following:
• The construction for a perpendicular bisector given above gives a clear procedure for drawing a
line, but it might not be clear why the line has to be the perpendicular bisector of the segment.
Use what you know from this class to prove that the line must be the perpendicular bisector.
• A common type of construction is to find a regular polygon. This is a convex polygon whose
side lengths are all equal and angles are all equal. It is relatively simply to both move angles
and segments using Euclidean constructions. Explain why this means the most complicated step
in constructing a regular polygon is to find one angle of the polygon. Since we are in Euclidean
geometry, what would be the measure of one angle of a regular n-gon.
• Find constructions for the regular triangle (equilateral triangle) and regular quadrilateral (square).
These are well known (in fact the construction of a regular triangle is the first proof in Euclid’s
Elements), so if you get stuck you should be able to find resources to help you out.
• Given an angle, it is possible to bisect it using Euclidean tools. (But generally not to trisect it!)
Using this fact, and your previous constructions, explain how you could construct an octagon or
hexagon.
• However, it is not possible to construct every regular polygon. Gauss in fact proved that it is
possible to construct a regular n-gon if n is a power of 2, or if it is the product of a power of 2
m
and distinct primes of the form Fm = 22 + 1 (the numbers Fm are called the Fermat Primes).
Using this result, which n-gons are constructible for 3 ≤ n ≤ 20?
• Try to see if you can find information about how Gauss proved his theorem. It is a deep result
with connections to Algebra, so I do not expect you to be able to prove it. But if you can, find a
sketch of the result and write a short discussion talking about your reaction to the proof.
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Area Calculations
In this problem you will investigate the difficulty of defining area, culminating in an indirect proof of
the area of a circle. Area is a somewhat tricky concept, though often the complications in defining area
are overlooked.
• We measure area in square units, such as square feet or square meters. This measurement is
actually a ratio. For example, we might say that a square with side length of two feet is four
square feet in size, because it is four times as big as a square with side length of one foot (which
we call the “square foot”). When we say that the square is “four times as large as the smaller one,
we mean that we can completely cover the larger square with four copies of the smaller square
without overlap. But if we have a triangle whose area is four square feet, we may not be able to
actually fit four squares into the triangle. Why then do we say that the area is four square feet?
• Expand upon the last question to try to answer what rational areas (such as (4/3) square feet)
and irrational areas (such as π square feet) mean in relationship to a square with unit side length.
Propose a set of basic properties that you think area should satisfy (for example, one such property
should be “congruent shapes have the same areas.”)
• Using your discussion of area in the last section, justify the formula for the area of a rectangle
(that is, that if the length is ` and the width is w, the area is ` · w). Be careful in your justification
to handle fractions and irrational numbers.
• From here, argue that the area of a right triangle must be (1/2)` · w, where ` and w are the lengths
of the legs (not the hypotenuse). You should be able to prove this from your formula for the area
of a rectangle together with your basic properties of area.
• From here, find a formula for the area of a regular n-gon (a regular n-gon is a convex n-gon whose
sides are all congruent and whose angles are all congruent). Again, you should be able to prove
your formula from your work in the previous sections.
• The area formula for a convex n-gon should suggest an area formula for the circle. Find it, and
state an intuitive argument for using your previous results to show that this formula must be
correct. (However, if you have done the previous parts correctly, you will have done a significant
amount of work by this point and so you do not need to prove this last result).
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The Pythagorean Theorem
One of the oldest and most famous achievements of Euclidean geometry is the Pythagorean theorem.
It has been said that this theorem has been proven in more ways than any other results in all of
mathematics. We will use similar triangles to prove this result in class. In this problem, you will
investigate alternative techniques.
• Take the following as given: “The area of a rectangle with side lengths ` and w is ` · w,” “The area
of a right triangle with legs of length ` and w is (1/2)` · w,” and “If a shape is divided into smaller
parts which overlap only on their boundaries, then the area of the whole is equal to the sum of
the area of the parts.” Using this, together with the results from class, prove the Pythagorean
theorem without using similar triangles. (Note that the Pythagorean theorem is only valid in
Euclidean geometry, so you will need to use results from Euclidean geometry).
• Now prove the theorem again using an alternative approach. (One approach is this: given a right
triangle whose legs are of length a and b, and whose hypotenuses is of length c, create a square
with side lengths a + b and draw four right triangles inside of it to create a square with side length
c. Calculate the area of the outer square in two ways: directly from the area formula and by
adding up the areas of the four triangles and the inner square. This will give you an equation
which should simplify to a2 + b2 = c2 . If you already used this approach in the previous section,
you will need to investigate a new approach, but there are many historical techniques which prove
the Pythagorean theorem).
• The Pythagorean theorem is not true in Hyperbolic or Spherical geometry. Examine the three
proofs you have of the theorem (the one from class and the two you proved in the previous parts
of this problem). Each one should use an assumption which is not true outside of Euclidean
geometry, identify it in every case (if you cannot find such an assumption, then your proof is
incomplete). Then write a brief discussion which guesses why the theorem might fail in the other
two geometries. In those geometries, would we get an inequality instead, or could the equation
sometimes hold and sometimes not hold, depending on the right triangle?
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Euclid’s Elements, and the Language of Modern Mathematics
In this problem you will investigate the most famous geometric work of all time, Euclid’s elements. You
can find a copy of Euclid’s elements online at http://aleph0.clarku.edu/ djoyce/java/elements/elements.html,
or for free at several other locations since it is in the public domain and even some its translations are
in the public domain.
For this problem, you must complete the following:
• Euclid starts Book I with a set of 23 definitions and no undefined terms. We started with lines and
points as undefined terms, and based other definitions upon these terms. Because Euclid tries to
define everything, several of his definitions would not be considered rigorous definitions in modern
mathematics. However, several of his definitions are perfectly acceptable. Find two definitions
that you think are rigorous and two definitions that you think are not rigorous, and explain what
you think is good and bad about each definition.
• Euclid continues with 5 postulates, which we would call axioms (though sometimes we also call
them postulates). Each of these postulates corresponds in some way to an axiom from class.
Identify each postulate with the axiom closest to it from class, explaining why you make your
choices. Euclid only has 5 postulates as compared to our 14 axioms for Euclidean geometry. Why
might Euclid have used fewer postulates? Is he missing some, or are we using too many?
• Euclid further continues with 5 “common notions” which are axioms which would be relevant to
disciplines beyond geometry (as opposed to his postulates, which are geometric in nature). His
common notions initially look like algebraic statements. However, they can be expressed in a more
geometric form. Describe how these common notions relate to geometric objects like segments
and angles.
• Read Euclid’s first proposition and its proof. Why do you think Euclid might have started from
this result? Is his proof rigorous from a modern viewpoint? Does it use any assumptions not
present in Euclid’s postulates or common notions?
• Finally, restate Euclid’s first proposition using modern language and prove it using the results
from our class.
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High School Approaches to Triangle Congruence
Some high school geometry classes omit what are considered to be “more difficult results.” This will
almost always include the SSA theorem, and sometimes will include the AAS theorem and the fact that
if D is in the interior of 4ABC, then the lengths DA and DB cannot be the same as CA and CB.
This forces a different approach to proving various results about triangles. In this problem, you will
explore what that might mean.
Throughout this problem you are free to assume α = ∞, as it would be in a High School class. This
will simplify some proofs.
For this problem, you must complete the following:
• The starting point for most geometry classes will be to discuss SAS and ASA congruence, isosceles
triangles and the exterior angle theorem. They will prove these things in much the same way that
we proved them in our class (without the technical considerations related to side lengths of the
triangle for the exterior angle theorem), so there is no need to prove these a second time for this
problem.
• In our class we first proved the triangle inequality, and then used this to prove the scalene inequality. Since the scalene inequality seems more intuitive to many students (since it relates the size
of an angle to the size of a line in the angle), many classes will first prove the Scalene inequality,
and then use that to prove the Triangle Inequality. Prove the Scalene inequality using only SAS
congruence, ASA congruence, the Isosceles triangle theorem and the exterior angle inequality, and
then prove the triangle inequality using the Scalene inequality together with the previous results.
• After this, many classes will prove the SAS inequality. They will do so in the same way that
we did in class, so there is no need to repeat this argument. However, write a short discussion
describing why the SAS inequality intuitively seems to be true. You should note why it is only
true if the side lengths are fixed as the angle varies.
• At this point, high school classes will often prove the SSS congruence condition using the SAS
inequality. Write a proof which does this.
• From here the AAS congruence would be proven like we did in class, but the SSA theorem is
usually not proven since it is not a congruence condition. Sometimes however, what is called
“SsA congruence” is proven. This is the statement that if we know BAC ∼
= EDF , AB ∼
= DE,
∼
∼
BC = EF and we know that BC > BA (so that EF > ED), then 4ABC = 4DEF . Prove this
using only the results from the previous parts of this problem. How is this result consistent with
our SSA theorem?
• Finally, write a brief discussion comparing the techniques you used to prove the congruence criteria
in this problem with the approach we used in class. Which do you find simpler? Which makes
more sense to you? Which do you find more satisfying?
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Pole-Equator Triangles on the Sphere
You may recall that a review problem from the first exam asked you to calculate the exact measurement
of the angles in a triangle with sides lengths α/2, α/2 and α/4 on a sphere. In this problem, you will
try to generalize this problem to calculate the angles of arbitrary triangles with one vertex of a pole
and the other two vertices on the equator.
All of the calculations in this problem will be done in spherical geometry (where α < ∞).
For this problem, you must complete the following:
• First prove that if a triangle has two sides of length α/2 and one side of length α/4, then it has
two right angles and one angle of exactly 45 degrees. (Or in other words, it has two angles of
measure β/2 and one angle of measure β/4).
• Using a technique similar to the above argument, determine what the angles must be in a triangle
with two sides of length α/2 and one side of length α/2n , where n is an integer greater than 1.
• From here, determine what the angles must be in a triangle with two sides of length α/2 and one
side of length α/r, where r is a dyadic rational number. (That is, r = p/2q where p and q are
nonnegative integers.) Based on your calculations for these triangles, conjecture what the angles
in a triangle with two sides of length α/2 and one side of length α/x, where x is a real number,
might be.
• Show that given any real number x, and any positive error ε, that there is a dyadic rational
number r so that |x − r| < ε. Explain why this suggests that your formula for the angles in
pole-equator triangles from the last part is probably correct for all real number divisors of α, not
just the dyadic rational divisors.
• Write a short discussion in which you describe a different way to find the angles in these triangles,
without using any triangle congruences whatsoever. (This will likely not be a proof).
• Finally, comment on your thoughts to whether this technique could be generalized to triangles
without sides touching the pole and the equator but still on the sphere. If we could find a way
to determine the angles of a triangle given only its side lengths, we would have something like
trigonometry on a sphere. Do you think that such a thing is possible and if so, how might you go
about developing it? (If not, what would prevent you from developing something like trigonometry
on a sphere)?
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Cartesian Geometry
You have probably become accustomed to describing shapes in your previous math classes by using
coordinates. For example, you may talk about a line as the set of points which satisfy the equation
y = mx + b or a circle as the set of points which satisfy x2 + y 2 = r2 . But where do these variables
x and y come from? If we simply look at a line or a circle, there is no obvious way to label the
points (or perhaps, there are too many ways to label them, since different people could label them in
different ways). Classical geometry did not use coordinates (and the field of Geometry called “Synthetic
Geometry” avoids coordinates). The current coordinate system was invented by the mathematician
Descartes, which is why they are sometimes called “Cartesian coordinates.” In this problem you will
investigate how Descartes invented coordinates.
For this problem, you must complete the following:
• Suppose that we are Euclidean geometry. Pick any line `. Call this line the “x-axis.” Pick any
point O on `. Call this point the “origin.” Find a line m which contains O and so that ` ⊥ m.
Call this line the “y-axis.” Choose coordinates on ` and m so that O has coordinate 0 on both
lines. Explain why we can do each of these operations.
• Now suppose that P is any other point in the plane. Find a way to associate it with a coordinate
x on the x-axis and a coordinate y on the y-axis. (In this part of the problem you will not be able
to use coordinates, since you are defining coordinates). After you have chosen a way to associate
a coordinate (x, y) with a point, answer the following questions: If we have a point P , is there
exactly one coordinate (x, y) which corresponds to it? If we have a coordinate (x, y), is there
exactly one point P which corresponds to it? (If you answer “no” to either question, you should
go back and pick coordinates in a different way).
• Now that you have a way of assigning coordinates to points in the plane, try to use them with
equations. Consider the set of points satisfying the equation y = mx + b for some constants m
and b. Show that this set is actually a line. (Remember, we are considering lines to exist before
we choose coordinates, or else we wouldn’t be able to find an “x-axis” or “y-axis” to determine
the coordinates in the first place).
• Finally, consider whether your coordinate system would work in Spherical geometry. If so, what
kind of coordinates would you get there? Would every point correspond to one coordinate? Also
consider the fact that we use the coordinates of “longitude” and “latitude” to indicate locations
on the Earth in real life. Are these coordinates the same as the ones that you found? Write a
short discussion talking about the use of coordinates on spheres.
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The Triangle Inequality and Length
We consider the triangle inequality important because it in some way tells us that “a line segment is
the shortest way between two points.” In this problem you will try investigate this statement in more
detail.
For this problem you must complete the following:
• First prove the polygonal path inequality. This is the statement: “if P1 , P2 , . . . , Pn are distinct
points, then:
P1 Pn < P1 P2 + P2 P3 + · · · + Pn−1 Pn
Explain what this has to do with polygonal paths.
• Now we will try to deal with general paths. The only “curved” path that we have is an arc of a
circle. So we will try to argue the following: “If A and B are points on a circle, then the distance
AB is strictly less than the length of either arc which connects A and B.” We will argue this
by saying that the length of an arc of a circle can be approximated by a polygonal path which
connects a large number of points on the arc. To test to see if this plausible, do the following:
Find the length of the perimeter of a regular n-gon inscribed inside of a circle of radius r. Examine
this calculation as n gets very large. Does it get close to a familiar calculation about the circle?
If so, it is at least plausible that we can approximate arc lengths by considering polygonal paths
which connect points on the arc.
• Taking it as given that the length of arc is very close to the length of a polygonal path which
connects a large number of points on the arc, argue that the line segment which connects the
endpoints of the arc must always have shorter length than the arc itself.
• Discuss how you would generalize this technique to general curves. How does this suggest that a
line segment is the shortest path between two points?
• Given the above, is it appropriate to say that if we get between two points by traveling along a
line that we have taken the shortest distance between two points? Discuss why this may be true
or not true generally (in both Euclidean and non-Euclidean geometry).
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Chords in Circles
We have seen that the further away a chord is from the center of the circle, the shorter the chord is. But
how exactly does the distance relate to the length? You will investigate this question in this problem.
For this problem you must complete the following:
• First study whether the relationship between the distance and length of a chord is simply an inverse
proportion. That is if we know that AB and CD are chords in a circle, and that AB = kCD, is
it true that the distance of CD to the center of the circle is k times that of AB?
• Unfortunately, to fully write a guideline for this project, I have to spoil the last question. You
should be able to figure out that no, the inverse proportional relation does not seem plausible.
Prove that it is not true. (You can simply prove one counterexample, such as by finding two
chords so that one is twice as far form the center as the other, but is not half the length of the
other chord). This may be easier to prove using theorems from Euclidean geometry.
• The previous parts show that the relation between the distance to the center of the circle and the
length of a chord is a bit more complicated. To start investigating the problem, suppose that you
have a circle of radius 1, and a chord of distance x away from the center. Find a formula for the
length of the chord, and prove that your formula is correct.
• Using your formula from the last part, is there something that we can say about the lengths of
two chords if one is twice as far away from the center as the other? If so, prove it. If not, explain
why we cannot say anything.
• You probably used results from Euclidean geometry in the previous parts to this problem. Do you
think that your results about the lengths of chords would also apply to Spherical and Hyperbolic
geometry? If not, what would change (for example, would the chords be longer or shorter in
Spherical geometry, or can’t we tell?) Write a brief discussion about this.
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The Foundations of Trigonometry
Trigonometry is a very useful tool for finding lengths and measures of angles in Euclidean geometry. In
this problem you will investigate how trigonometry can be developed from geometry.
For this problem you must complete the following:
• We want to define the trigonometric functions (sin, cos, tan, sec, csc, cot). These functions will
allow us to calculate the side lengths of a right triangle, given knowledge of one angle and one
side of a right triangle, and their inverses will allow us to calculate the angles of a right triangle,
given that we know two of their side lengths. For this even to be possible to attempt, we need
to have the following be true: “if two right triangles have a pair of congruent sides and a pair of
congruent angles, then they are congruent” and “if two right triangles have two pair of congruent
sides, then they are congruent.” Figure out whether these statements are true only in Euclidean
geometry, or also in non-Euclidean geometry, and prove them where it is possible.
• Our trigonometric functions calculate side lengths in a special way. Given a right triangle and
knowledge of an angle, a trigonometric function tells us the ratio of two sides. For this to work
we need to have the following be true “given two right triangles which share an additional angle,
the ratio of two sides in one right triangle will always be equal to the ratio of the corresponding
sides in the other right triangle.” Is this true in non-Euclidean geometry, or only in Euclidean
geometry? Prove it where you can.
• With these facts it is possible to define the trigonometric functions. But why did each trigonometric
function receive it’s name? We can decide names for four of the trigonometric functions by drawing
the following picture: Given a right triangle 4ABC with right angle ∠ACB so that AC = 1,
let ∠ABC = θ. Draw a circle with center A and radius 1, then reflect 4ABC over the line
←→
AB. You should be able to find segments in your diagram whose lengths correspond to tan(θ),
sec(θ), cot(θ), and csc(θ). Identify these segments and explain why the names “tangent,” “secant,”
“cotangent” and “cosecant” are appropriate for these segments. (sin and cos are derived from a
different diagram).
• Our trigonometric functions are only useful if we can find actual values of the function. Explain
how we can find the values of the trigonometric functions for 30, 45 and 60 degrees.
• From here, we can find additional values of the trigonometric functions. Explain how if we could
find sum, and difference rules for the trigonometric functions, we could use the previous known
values to find several more values for the trigonometric functions. Can we find the value of the
trigonometric angles in this way? If not, how would you suggest finding the remaining values
(by thinking about this, you will start to appreciate the careful work done in trigonometry before
there were computers or tables of trigonometric values).
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Linkages
In this problem you will study the concept of linkages. Mathematical linkages are meant to simulate
constructions where rods are joined together at their ends. The ends themselves may or may not be
allowed to move. For example the “double pendulum” problem from physics studies a linkages of two
rods joined together at one end, where one of the rods (the top pendulum) has its end fixed to a point.
The other ends are allowed to move, which allows the whole system to rotate.
Throughout this problem you may work in Euclidean geometry.
For this problem, you must complete the following:
• Generally we will describe a linkage by describing it as a set of polygons whose lengths are known
(since the sides correspond to rods which cannot expand or shrink) but whose angles are unknown.
If there is more than one possibility for an angle, we will say that the linkage can move, but if
there is only one possibility for all angles then the linkage cannot move. For example, consider the
linkage of the three rods AB, BC and AC which are joined in the triangle 4ABC and all have
length 1. Show that in this situation, there is only one possibility for the angles in the triangle.
• Now consider an alternative linkage, formed by the rods W X, XY , Y Z and W Z all of length 1
which are joined in the quadrilateral ♦W XY Z. Show that there are many possibilities for the
angles in ♦W XY Z, but that it is not possible to find any combination of four angles.
• Using your results from the previous two parts, write a short discussion explaining why triangles
are considered to be a “stable” shape in architecture, but quadrilaterals must be braced in order
to be stable.
• Now you will investigate a more complicated linkage (known as the Peaucillear linkage). Let AB,
BC, CD, DA, XA, XC and OB be rods so that ♦ABCD is a rhombus, XA ∼
= XC, so that the
distance from O to X is the same as OB and so that the points O and X cannot move. (Note
that this means that point B is free to move along a circle of radius OB with center O and so
that X is another point of the circle). First prove that no matter what the configuration of the
linkage, that the points X, B and D are collinear.
• Using your work from the last part, show that in any configuration of the Peaucillear linkage, the
product OB · OD is the same. (Hint: Let P be the intersection of AC and BD. Show that this
divides ♦ABCD into four right triangles, and use properties of triangles to get the desired result.)
• Finally, write a short discussion explaining what the use of the Peaucillear linkage might be. (If
you have no idea, you should be able to find information about the history of this linkage, as it is
quite famous.)
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Your Suggestion
If you do not like any of the previous problems, you can suggest your own. Keep in mind that I have veto
power over any of your suggestions, and may only agree to a problem after I make several modifications
to your original suggestion. After I have made any such modifications, you will be free to choose a
different problem if you no longer like your problem.
Here are some problems that I considered, but was not able to write a complete write-up for:
• A study of how to find lines, distances and angles on a sphere by using three dimensional coordinates.
• A study of the various models for the hyperbolic plane.
• A project which involves creating constructions in a computer visualization system, like Cinderella.
• A project which relates continuity from Calculus to our notion that lines and circles should not
have gaps, as well as notions such as the fact that distance from a point should increase or decrease
continuously as we move along a line.
• Projects discussing famous results from geometry, such as Heron’s formula for the area of a triangle.
• A problem which explains what “squaring the circle” is and why it is a synonym for doing something impossible.
• A problem about the various centers of a triangle, the Euler line which connects them, or the
9-point circle which contains many points of interest on a triangle.
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