MATH 371 Assignments Spring, 2003

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TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE with  HSU ONCORE
The Elements by Euclid, 3 volumes, edited by T.L. Heath, Dover (1926)
Proof in Geometry by A.I Fetisov, Mir (1978)
Here's Looking at Euclid..., by J.Petit, Kaufmann (1985).
Flatland By E. Abbott, Dover.

Tentative assignmentsand topics for classes. 3/30/03 Subject to Revisions
Week Monday Wednesday Friday Reading for the week. Problems 
Due on Wednesday 
of the next week
1 1/20 No Class 1/23 1.1 Beginnings 
What is Geometry? 
The Pythagorean Theorem 
1/25Intro to Geometer's Sketchpad/ Wingeom 


M&I:1.1, 1.2
E:I Def'ns, etc. p153-5; 
Prop. 1-12,22,23,47 
A:.Complete in three weeks
M&I p5:1-8,11 Due: 1/29 
2 1/27 The Pythagorean Theorem  1/29 
1.2 Equidecomposable Polygons
1/31 M&I 1.2, 1.3 
E: I Prop. 16, 27-32, 35-45.
M&I: p10:1,2,5,10,11-13 Due:2/6 
Prove:The line segment connecting the midpoints of  two sides of a triangle is parallel to the third side and is congruent to one half of the third side. 
Lab Exercises 1: Due: 1/31
Construct a sketch with technology of 
1. Euclid's Proposition 1 in Book I. 
2. Euclid's Proposition 2 in Book I. 
3. One "proof" of the Pythagorean Theorem. 
3 2/3 More on Constructions 
1.1 Def'ns- Objects 
1.2 Constructions 
2/5 1.3 Geometry and numbers  2/7 1.4 Continuity M&I 1.3,1.4 
E: III Prop. 1-3, 14-18, 20, 21, 10 
F. Sect. 11, 25, 31
M&I: p17:5, 8-11 
p11: 16-19, 24, *27  Due:2/13 
Problem  Set 1 Due:2/13 
Lab Exercises 2: Due by 2/8.
Do Construction 3, 4, 6, 7, and 8 from Meserve and Izzo Section 1.2.
BONUS:Show how to "add" two arbitrary triangles to create a single parallelogram.
4 2/10 Transformations - Isometries 2/12 Coordinates and Transformations  2/14 Inversion 
Begin Affine Geometry
M&I:1.5, 1.6, 2.1 
E: V def'ns 1-7;VI: prop 1&2 
F. Sect. 32
M&I:1.6:1-12,17,18 Due 2/20 
Problem Set 2 Due 2/20 
Lab Exercises 3: Due 2/14. 
1. Construct a scalene triangle using Wingeometry. Illustrate how to do i) a translation by a given "vector", ii) a rotation by a given angle measure, and iii) reflections across a given line..
2. Create a sketch that shows that the product of two reflections is either a translation or a rotation.
5 2/17 Affine Geometry 2/19 Affine Geometry 2/21 More affine geometry.  M&I: 2.1, 2.2,
E:IV Prop. 3-5
M&I: p23: 9,10 (analytic proofs) Due 2/27 
Lab Exercises 4: Due 2/21.
1. Draw a figure showing the product of three planar reflections as a glide reflection.
2. Draw a figure illustrating the effects of a central similarity on a triangle using magnification or dilation that is a) positive number >1, b) a positive number <1, and c) a negative number.
6 2/24 Orthogonal circles and Inversion. 2/26 More Inversion and Affine Geometry (planar coordinates) 2/28 Seeing the infinite  M&I: 2.1,2.2 Problem Set 3 (Isos Tri) Due 3/6  [4 Points for every distinct correct proof of any of these problems.] 
Lab Exercises 5: Due 2/28.
Construct the inverse of a point with respect to a circle a) when the point is inside the circle; b) when the point is outside the circle.
7 3/3 Homogeneous Coordinates. 3/5More on Affine geometry- Homogeneous coordinates and visiaulizing the affine plane. 3/7 More on Homoogeneous coordinates for the plane. M&I: 3.1,3.2, 3.5  Due 3/13  M&I: 3.5: 1,3,4,5,10,11 
Lab Exercise 6: Due 3/7 . See Notes [Abridged]
1.Draw sketches for each of the following triangle coincidences:
1. Medians. 2. Angle Bisectors. 3. Altitudes. 4. Perpendicular Bisectors.
2.  Given a circle O and two interior points A and B, construct an orthogonal circle O' through A and B. 
3. Draw two intersecting circles O and O' and measure the angle between them. 
8 3/10 Begin Synthetic Geometry [Finite] 
Homogeneous Coordinates with Z2 and Z3
3/12 More on Finite Synthetic Geometry and models.  3/14 Algebraic-projective geometry: Points and lines. M&I:3.6, 3.4,3.7 Lab Exercise 7: Due 3/14 . 
1. Inversion: Investigate and sketch the result of inversion on lines and circles in the plane with a given circle for inversion. 
When does a line invert to a line? When does a line invert to a circle? When does a circle invert to a line? when does a circle invert to a circle?  Show sketches where each case occurs. [ Remember the inverse of the inverse is the original figure.] 
2. Use inversion with respect to the circle OP to invert <BAC to <B'A'C'. Discuss briefly the effects of inversion on angles. 
3. Draw a sketch of the affine plane showing the horizon line and label the lines X=1,2,-1, Y= 1,2,-1 and points (1,2) and (2,-1).
Due: 3/27
Problem Set 4
3.6: 3,7-15 
3.7: 1,4,7,10,13 
9 Spring break 3/17 No Class 3/19 No Class 3/21
10 3/24 Spatial and  Planar Desargues' Theorem 
Begin Synthetic Projective Geometry 
3/26 Projective Geometry -Visual/algebraic and Synthetic... Axioms 1-6  3/28 -Projective Planes. 
Triangle Coincidences (Perpendicular Bisectors- the circumcenter) 
M&I:4.1, 4.2, 4.3, 2.4 Lab Exercise 8: Due 3/28
Draw a sketch for Desargue's theorem in the plane.
Due 4/10. (changed 4-2)
Prove P6 for RP(2); 
4.2: 2,3, Supp:1 
4.3: 1-6, Supp:1,5,6
11 3/31No Class.
CC Day.
4/2 More on the axioms of Projective Geometry. 
Proof of Desargues' Theorem
Duality Theorem and Desargues. 
M&I:4.1, 4.2, 4.3, 2.4 No Lab this week. 
12 4/7 Conic Sections.
Pascal and More Duality
4/9 Complete quadrangles Postulate 9. 
Projective transformations. 
4/11 Perspectivities and Projectivities. 
Conics ? 
Pascal's Theorem ?
4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm)
Lab Exercise 9: Due 4/11
Pascal's configuration: Hexagons inscribed in conics. Points of intersections of opposite sides lie on a single line. 
Construct a figure for Pascal's configuration  with  a) an ellipse , b)a parabola,  and c) an hyperbola. 
13 4/14 More on coordinates and transformations.  4/16 Projectivities. Perspective  4/18Transformations of lines with homogeneous coordinates. 4.10, 5.4, 2.4 
Lab Exercise 10: Due 4/18
1. Construct a sketch showing ABC on a line perspectively related to A'B'C' on a second line with center O. 1'. Draw a dual sketch for the figure in problem 1. 
2. Construct a sketch of  ABC on a line projectively (but not perspectively) related to A'B'C' on a second line. Show two centers and an intermediate line that gives the projectivity. 2'. Draw a dual sketch for the figure in problem 2. 
3.Construct a sketch of  ABC on a line projectively (but not perspectively) related to A'B'C' on the same line. Show two centers and an intermediate line that gives the projectivity. 2'. Draw a dual sketch for the figure in problem 3 
Due : 4/17
M&I:4.5:2; 4.6:7,8,9; 4.7:4,7 
[Prove P9 for RP(2),optional]
14 4/21 Projectivities in 3 space: More on Projective Line Transformations with Coordinates. Begin Harmonic sets  4/23 Harmonics: uniqueness and construction of coordinates for a Projective Line, Plane, Space.  4/25  A Non-Euclidean Universe. 5.1,5.4 Lab Exercise 11: Due May 2
1.Draw a sketch showing H(AB,CD) and H(CD, AB).
2. Draw a sketch that shows that if H(AB,CD) and H(AB,CD*) then D= D*. 
3.(new) Use five points and Pascal's Theorem to constuct a conic.
Due :5/1 M&I: 4.10:1,3,6,7; 
5.1:5; 5.4:1-8,10;  5.5: 2,3,7
15 4/28 Projective genration of conics. 
More on Transformations, Coordinates and Harmonic sets.
Matrices for familiar Planar Projective Transformations. 
4/30   5/2 Conics revisited. Pascal's and Brianchon's Theorems. Equations for conics. 5.1,5.2, 5.3,5.5, 5.7, 6.1, 6.2
16 5/5 The Big picture in Summary. 
Inversion properties?
5/7 Student Presentations 5/9 6.4, 6.6, 6.7

Problem Set 1

DEFINITIONS: A figure C is called convex if for any two points in the figure, the line segment determined by those two points is also contained in the figure.
That is, if A is a point of C and B is a point of C then the line segment AB is a subset of C.

If F and G are figures then F int G  is { X : X in F and X  in G }.
F int G is called the intersection of F and G.
If A is a family of figures (possibly infinite), then  int A = { X : for every figure F in the family A, X  is in F }.
int A is called the intersection of the family A.

1. Prove: If F and G are convex figures , then F int G is a convex figure.

2. Give a counterexample for the converse of problem 1.

3. Prove: If A is a family of convex figures, then int A is a convex figure.

4. Prove: The line segment RS is convex. [ Refer to M & I pg.2.] 

Problem Set 2

1. Suppose n is a natural number. Given P0 and P1 , prove by induction that you can construct with straight edge and compass (SEC) a point P sqrt(n)   which will correspond to the number  sqrt(n) on a Euclidean line.

2. Suppose we are given P0, P1, and P a where P a corresponds to the real number a>0. Give a construction with SEC of a point Psqrt(a) which will correspond to the number  sqrt(a) on a Euclidean line.

3. Given points P0, P1, Px, and Py on a Euclidean line corresponding to the real numbers x>0 and y>0, give constructions with SEC for the following points.

a) P x + y b) P x - y c) P x *y d) P 1/x
4. Construct with SEC on a Euclidean line:  sqrt( sqrt(5)/sqrt(3)  + sqrt(6) ).

5. Suppose that d(A,B) = d(A',B') and that l is the perpendicular bisector of the line segment  AA'. Let B'' be the reflection of B across l, i.e., B''= Tl(B). Prove that if  B' is not equal to B''  then A' lies on the perpendicular bisector of the line segment.

Problem Set 3

1.  Prove: Two of the medians of an isosceles triangle are congruent.

2.  Prove: If two of the medians of a triangle are congruent then the triangle is isosceles.

3.  Prove: The angle bisectors of congruent angles of an isosceles triangle are congruent.

4.  Prove:  If two of the angle bisectors of a triangle are congruent then the triangle is isosceles. 

Problem Set  4

1. Use an affine line with P0 , P1 , and Pinf given. Show a construction for P1/2 and P2/3.

2. Use an affine line with P0 , P1 , and Pinf   given. Suppose x > 1.
Show a construction for  Px2 and Px3 when Px is known.

3. D is a circle with center N tangent to a line l at the point O and C is a circle that passes through the N and is tangent to l at O as well.
Suppose P is on l and PN intersect C = {Q}; Q' is on C so that Q'Q is parallel to ON; and {P'} = NQ'  intersect l.

Prove: a) P and Q are inverses with respect to the circle D.
b) P' and Q' are inverses with respect to the circle D.
c) P and P' are inverses with respect to the circle with center at O and radius ON.

4. Suppose C is a circle with center O and D is a circle with O  an element of D.
Let I be the inversion transformation with respect to C.

Prove: There is a line l, where I(P) is an element of  l for all P  that are elements of  D -{O}.