Homework 1, due Tuesday 8/21

In class today we discussed four different centers of a triangle:

circumcenter
centroid
incenter
orthocenter

Write a report on two of these centers. (Your report should be a total of four pages.)
For each of the centers you choose, explain how to construct it using GSP, and explain
any special properties it has.

Do you notice any surprises (theorems) about your centers? For example, it's surprising
that the three angle bisectors of a triangle all go through the same point. Can you prove
this is true?

Discuss the relation between the two centers you choose. For example, for which
triangles are the two centers the same?

Are both of your centers always inside the triangle? If not, for which triangles are one
or both of your centers outside the triangle?

Can you find any interesting properties of your centers involving distance, angles, or
area? For example, what's the relation among the lengths of the six segments formed
when you intersect each angle bisector with the opposite side of the triangle?

Can you think of any other types of centers of a triangle?

Homework 2, due Tuesday 8/28

1. Let ABC be a right triangle, and suppose that DBC, AEC, ABF are equilateral triangles.
What is the relation between the areas of these three equilateral triangles? Turn in a
GSP experiment which verifies your answer. Extra credit: Prove your answer.

2. Let ABC be a triangle. Let D be a point on the segment BC, let E be a point on the
segment AC, and let F be a point on the segment AB, so that the lines AD, BE, CF are
concurrent. Find a relation between the lengths of the six segments BD, CD, AE, CE,
AF, BF. Turn in a GSP experiment which verifies your answer. (You do not have to prove
your answer.)

clue to above problem

Let ABC be a triangle. Let D be a point on the segment BC, let E be a point
on the segment AC, and let F be a point on the segment AB, so that the lines
AD, BE, CF are concurrent. Consider the lengths of the following six
segments listed IN ORDER around the boundary of the triangle: AE, EC, CD,
DB, BF, FA. Find a relation between the three numbers AE/EC, CD/DB, BF/FA.
Turn in a GSP experiment which verifies your answer. (You do not have to
prove your answer.)

3. Use GSP to construct regular polygons with n sides for each of the following values: n
= 3, 4, 6, 8, 12. In each case your given data should be two points, which will be the
endpoints of a side of the polygon. You should also give a written description of your
construction, step by step. (You do not have to prove that your construction works,
however.)

4. State and prove the Pythagorean theorem. (For this problem it is OK to look up a
proof in a book or on the internet.) Use GSP to illustrate your proof. Be prepared to
present your proof to the class.

5. If x and y are positive real numbers, the geometric mean m of x and y is the square
root of xy. In other words, m2 = xy, or x/m = m/y. Because of this last equation, m is
also called the "mean proportional" of x and y.

(a) Prove that the altitude on the hypotenuse of a right triangle is the geometric mean
of the segments of the hypotenuse. In other words, if ABC is a triangle with right angle
at vertex C, and CD is the altitude from C to the side AB, then CD is the geometric
mean of AD and DB. Write your proof in complete sentences.

(b) Given a triangle, how do you construct a square with the same area? Use part (a)!
Give a careful step by step explanation using the GSP Construct menu. Explain why your
construction works.

Note: For this homework, you should turn in your GSP files in electronic form. You may
turn in a floppy disk, or you may email the files to clint@math.uga.edu (as attachments).
You may also turn in the written part of the assignment electronically. Emailed
homework must be sent no later than the beginning of class, 11:00 am on 8/28/01.

Return to Math 7200

Homework 3, due Tuesday 9/4

The solution to each problem should include a GSP sketch (in electronic form).
In problems 1 and 2, "construct" means to construct using only the Greek construction rules and the GSP
construction rules which can be derived from them, i.e. all of the GSP Construct menu except Point on Object
and Locus.
In problems 1 and 2 give a step-by-step description of each construction.

1. Following the outline given in class, construct a regular pentagon, starting with a side AB of the pentagon.

2.(a) Given points A and B, show how to construct two points C and D which divide the line segment AB into 3
segments of equal length.
(b) Explain how to do a construction which divides the line segment AB into n segments of equal length (n > 1).
Illustrate your construction for the case n = 7.
(c) Prove that your construction works!

3. Quadrilaterals.
(a) Suppose that ABCD is a quadrilateral. Let P, Q, R, S be the midpoints of AB, BC, CD, DA, respectively. What can
you say about the quadrilateral PQRS? What is the relation between the area of PQRS and the area of ABCD?
(b) Prove your assertions in part (a).

4. Areas of parallelograms.
(a) Draw a parallelogram, and label the vertices A, B, C, D (in order around the figure).
(b) Construct the diagonal AC.
(c) Pick a point P on AC. (Use the command Point on Object in the Construct menu.)
(d) Construct lines through P parallel to the two sides of the parallelogram ABCD.
(e) Measure the areas of the four smaller parallelograms into which ABCD is divided by the lines you just
constructed. What do you see?
(f) Extra credit: Prove your observation in part (e).

5. A triangle construction.
(a) Draw a triangle, and label the vertices A, B, C.
(b) Construct the midpoints of the three sides, and label them L, M, N.
(c) Construct the feet of the altitudes of the triangle ABC, and label them D, E, F.
(d) Construct the orthocenter H and the midpoints of the segments AH, BH, CH, and label them X, Y, Z. What do you
see?

MATH 5200/7200 Homework 4

Don't hand in. These problems will be discussed in class on Tuesday 9/11.

Write your proofs in complete sentences. Do not write "two-column proofs" except possibly as preliminary outlines of your final proofs.
You may use any basic facts you remember from plane geometry. If you are not sure of your facts ask a classmate. Look up answers only as a last
resort - these problems are meant to help you improve your proof skills.
Try to use GSP as an aid to discovering proofs.
In the construction problems use only the Greek construction rules. You may also use the items in the GSP Construct menu that can be done
using the Greek construction rules.

1. Prove: The angle bisectors of a triangle are concurrent.

2. Prove: If a triangle is inscribed in a circle so that one side of the triangle is a diameter of the circle, then the angle opposite that side is a right angle.

3. Prove: Let P, Q, R, S be points on the circle C, with P not equal to Q and R not equal to S. Let A be the minor arc of P and Q, and let B be the minor
arc of R and S. The central angles of A and B are equal if and only if the chords PQ and RS have the same length.

4. Prove: In a regular pentagon the ratio of a diagonal to a side is the golden ratio.
Outline of solution.

5. Given a triangle ABC, construct a similar triangle A'B'C' so that the area of A'B'C' is twice the area of ABC.

6. Given a line L, a point A, and a line segment BC, construct a circle X so that L is tangent to X, A lies on X, and X has radius of length BC. (Hint:
Given L and r, which circles of radius r have L as a tangent line?)

MATH 5200/7200 Homework 5

Due Tuesday 9/25

Homework groups were formed in class 9/20. Each group should turn in only one homework write-up. Of course all members of the group should
contribute to the write-up. Any member of the group may be called on in class to present the group's problem solutions.

1. Distance in Cartesian geometry. If point A has coordinates (x1,y1) and point B has coordinates (x2,y2), the distance from A to B is

d(A,B) = sqrt[(x1-x2)2 + (y1-y2)2].

Prove that this distance function satisfies the distance axiom. (Hint: Vectors are useful here.)

2. Distance in Minkowski geometry. If point A has coordinates (x1,y1) and point B has coordinates (x2,y2), the distance from A to B is

d(A,B) = |x1-x2| + |y1-y2|.

(a) Prove that this distance function satisfies the distance axiom.

(b) What are circles in Minkowski geometry? Explain how to construct the Minkowski circle with center A and radius r using the GSP Construct menu.

(c) If A and B are points, the "equal distance set" of A and B is the set of all points C such that d(C,A) = d(C,B). What are equal distance sets in
Minkowski geometry? Explain how to construct the Minkowski equal distance set of A and B using the GSP Construct menu. Equal distance sets may
look very different, depending on the position of the points A and B.

In your GSP constructions, you may take as "givens" the two coordinate axes.

MATH 5200/7200 Homework 7 - Don't hand in.

First here are some definitions involving circles.

Circle, center, radius: Let O be a point and let r be a positive real number. The circle C with center O and radius r is the set of all points P such that
d(O,P) = r.

Chord, radius, diameter: A chord of a circle C is a segment with both endpoints on C. A radius of a circle C is a segment with one endpoint the
center of C and the other endpoint on C. A diameter of C is a chord of C such that the center of C lies on the chord.

Secant, tangent: A secant of a circle C is a line S such that S meets C at exactly two points. A tangent of C is a line T such that T meets C at exactly
one point.

Arcs: An arc of a circle with endpoints P and Q is a major arc, a minor arc, or a semicircle. Let C be a circle with center O. Let P and Q be distinct
points on C, such that a(POQ) is not 180. The minor arc with endpoints P and Q is the set of all points X on C such that X = P or X = Q or ray(OX) is
between ray(OP) and ray(OQ). The major arc with endpoints P and Q is is the set of all points X on C such that X = P or X = Q or ray(OX) is not
between ray(OP) and ray(OQ). If a(POQ) = 180, the two semicircles with endpoints P and Q are the sets of all points X on C such that (1) X = P or X =
Q or a'(POX) > 0, (2) X = P or X = Q or a'(POX) < 0.

Directed arc: A directed arc of a circle is an arc together with an ordering of its endpoints.

Central angle of a directed arc: Let C be a circle with center O. If A is a directed arc of C, with ordered endpoints (P,Q), the central angle of A is the
angle POQ.

Inscribed angle: If the points P, Q, R are on the circle C, the angle PQR is inscribed in the circle C. The intercepted arc of the inscribed angle PQR is
the set of all points X on C such that ray(QX) is between ray(QP) and ray(QR).

Now here are some problems about circles.

Prove the following theorems, using only our axioms and our basic theorems. (We already discussed some of these problems before the axioms were
introduced.)

1. If P, Q, R are noncollinear, there is a unique circle C passing through P, Q, and R.

2. If P, Q, R are noncollinear, there is a unique circle C tangent to the lines PQ, QR, and PR.

3. Let P, Q, R, S be points on the circle C, with P not equal to Q and R not equal to S. Let A be the minor arc of P and Q, and let B be the minor arc of R
and S. The central angles of A and B have equal or opposite measures if and only if the chords PQ and RS have the same length.

4. The measure of an inscribed angle is equal to half the measure of the central angle of its intercepted arc.

5. Every trapezoid inscribed in a circle is isosceles.

6. An equilateral polygon inscribed in a circle is a regular polygon.

7. The radius of a circle inscribed in an equilateral triangle is equal to one-half the radius of the circumscribed circle and to one-third the altitude of the
triangle.

Homework 8, due Tuessday 10/23

Write careful proofs of the following theorems. You may use our axioms, our basic theorems, and any theorems already proved on homework or in class.
When you use an axiom or theorem, make an explicit reference to it.

If you cannot find a complete proof, here are some other things you can do:

Make a GSP construction which shows experimentally that the statement is true.
Prove a special case of the theorem.
Discuss your attempts to prove the theorem.

1. Opposite angles of a quadrilateral are supplementary if and only if the quadrilateral can be inscribed in a circle. (Half of this theorem was used in exam
2.)

2. The medians of a triangle are concurrent. (The idea of this proof has been discussed in class.)

3. (Extra Credit) Suppose that the points A, B, C are noncollinear, and the points A and A' are distinct. If AB is parallel to A'B', BC is parallel to B'C',
and AC is parallel to A'C', then the lines AA', BB', CC' are either concurrent or parallel.

MATH 5200/7200

Homework 9, due Tuesday 11/6

1. Choose one of the following three trig formulas: the area formula, the law of cosines, or the law of sines.

(a) Check it using GSP. (Turn in your GSP sketch on a disk or by email.)

(b) Prove it using the definitions, axioms, and theorems we have discussed in this course.

(c) Find an application or a generalization, and discuss it.

(You may discuss more than one formula for extra credit.)

The formulas refer to the triangle ABC, with sides a = BC, b = AC, c = AB. Following the usual shorthand notation, angles will be labelled by
their vertices.

I. Area formula:

Area = 1/2 ab sin(C)

Note that when C is a right angle, then sin(C) = 1, so this formula says Area = 1/2 ab, or 1/2 base times height. A possible application is to find
a formula for the area of a quadrilateral.

II. Law of cosines:

c2 = a2 + b2 - 2ab cos(C).

Note that when C is a right angle, then cos(C) = 0, so this formula says c2 = a2 + b2, which is just the Pythagorean theorem. A possible topic
for discussion is to interpret this formula geometrically, generalizing the interpretation of the Pythagorean theorem using squares on the three
sides of the triangle. Another might be a formula for the length of a side of a quadrilateral, in terms of the other three sides and the two angles
determined by these three sides.

III. Law of sines:

a/sin(A) = b/sin(B) = c/sin(C).

Note that when C is a right angle, then these equations just say that a/sin(A) = b/sin(B) = c, or sin(A) = a/c and sin(B) = b/c, which are just the
definitions of sin(A) and sin(B). A possible topic for discussion is that these numbers are equal to twice the circumradius of the triangle ABC.

For the following three proofs you may use axioms and theorems from this course, including the formulas from problem 1.

2. Prove that for a triangle ABC labelled as in problem 1, with circumradius R, the area of ABC is equal to abc/4R.

3. Let ABC be a triangle with circumradius R. Suppose that C1 and C2 are circles such that both C1 and C2 pass through the point A, C1 is
tangent to BC at B, and C2 is tangent to BC at C. Let p be the radius of C1 and let q be the radius of C2. Prove that pq = R2.

4. Prove that for a triangle ABC labelled as in problem 1, if D is the midpoint of side AB, then the length m of the median CD satisfies the
equation

m2 = 1/2 a2 + 1/2 b2 - 1/4 c2.

MATH 5200/7200

Homework 10, due 11/13

Prove the following theorems.

1. Suppose that two lines through a point P are tangent to a circle at points A and B. If the circle has center O and radius R, and Q is the
intersection point of the lines OP and AB, then (OP)(OQ) = R2.

2. If two lines through a point P meet a circle at points A, A' and B, B', respectively, then (PA)(PA') = (PB)(PB'). (Explanation: One line
meets the circle at two points A and A', and the other line meets the circle at two points B and B'.)

3. The feet of the perpendiculars from a point to the sides of a triangle are collinear if and only if the point lies on the circumcircle.
(Explanation: The foot of the perpendicular from a point to a line is the intersection of the line with the perpendicular line through the point.)