MATH 371 Assignments Spring, 2004


Back to Martin Flashman's Home Page :)   Last updated: 1/22/04

TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE with  HSU ONCORE through Blackboard.
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. 1/22/04 Blue cells Subject to Revisions
Week Monday Wednesday Friday Reading/Videos for the week. Problems 
Due on Wednesday 
of the next week		Math 480 Lab Assignments
1 1/19 No Class 1/21 1.1 Beginnings 

Lab: Intro to Geometer's Sketchpad/ Wingeom 

1/23
What is Geometry? 
The Pythagorean Theorem 
Transformations

 M&I:1.1, 1.2
E:I Def'ns, etc. p153-5; 
Prop. 1-12,22,23,47 
A:.Complete in three weeks		 Due: 1/28
M&I p5:1-8,11

2
 1/26 The Pythagorean Theorem 
 1/28
1.2 Equidecomposable Polygons
 1/30Finish proof of EP.
More on Constructions 
Isometries 
1.1 Def'ns- Objects 
1.2 Constructions
 M&I 1.2, 1.3 
E: I Prop. 16, 27-32, 35-45.
 Due: 2/4 M&I: p10:1,2,5,10,11-13
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: 2/3
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 2/2  1.3 Geometry: Constructions and numbers  2/41.4 Continuity 2/6  Beather:
Start: Transformations - Isometries		 M&I 1.3,1.4 
E: III Prop. 1-3, 14-18, 20, 21, 10 
F. Sect. 11, 25, 31
Watch Equidecomposable  Polygons
on -line Notes from Prof. Biles on trig.
 Due: 2/11  Changed
M&I: p17:5, 8-11 
p11: 16-19, 24, *27  Due:2/11changed
Problem  Set 1
REVISED: Now Due  2/16
No reading report for 2/16!
Lab Exercises 2: Due by 2/10.
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 square.
4 2/9 Inversion, Orthogonal Circles 2/11 Odds and ends. 2/13
Transformations - Isometries. Coordinates.		 M&I:1.5, 1.6, 2.1 
E: V def'ns 1-7;VI: prop 1&2 
F. Sect. 32		 Due: 2/11  Changed
M&I: p17:5, 8-11 
p11: 16-19, 24, *27  Due:2/11changed
Problem  Set 1
REVISED: Now Due  2/16
No reading report for 2/16!		 Lab Exercises 3: Due 2/17. 
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
 2/16 More Isometries: Coordinates and Transformations  

2/18 Isometries / coordinates/ classification
 2/20 Finish Classification of Planar Isometries.
 M&I: 2.1, 2.2,
E:IV Prop. 3-5
 Due 2/25:
M&I: p23: 9,10 (analytic proofs)
M&I:1.6:1-12,17,18 
Problem Set 2
Lab Exercises 4: Due 2/24.
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 2/23 Isometries and symmetries .... Begin Affine Geometry 2/25 Inversion and Affine Geometry (planar coordinates) 2/27 Seeing the infinite
M&I: 2.1,2.2 Due 3/5
Problem Set 3 (Isos Tri)   [4 Points for every distinct correct proof of any of these problems.]  		Lab Exercises 5: Due 3/2 .
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 3/1 More on the Affine Line. 3/3 More on Affine geometry- Homogeneous coordinates and visualizing the affine plane. 3/5Visualizing the affine plane. M&I: 3.1,3.2, 3.5 

Due 3/10
 M&I: 3.5: 1,3,4,5,10,11 

Lab Exercise 6:  Due 3/9
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
3/8  More on Homogeneous coordinates for the plane. 3/10 Begin Synthetic Geometry [Finite]  3/12 Homogeneous Coordinates with Z2 and Z3
More on Finite Synthetic Geometry and models.  		M&I:3.6, 3.4,3.7 Due: 3/24
Problem Set 4
3.6: 3,7-15 
3.7: 1,4,7,10,13		Lab Exercise 7: Due 3/23   . 
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).
9 Spring break 3/15 No Class 3/17 No Class 3/19 No Class


10 3/22 Axioms, consistency, completeness and models.
A non-euclidean universe.
 3/24 Algebraic-projective geometry: Points and lines.
Spatial and  Planar Desargues' Theorem
Proof of Desargues' Theorem
Begin Synthetic Projective Geometry 		 3/26 -Projective Geometry -Visual/algebraic and Synthetic..Axioms 1-6 Projective Planes. .  M&I:4.1, 4.2, 4.3, 2.4
Due 4/7.
M&I:4.1:7,15,16; 
Prove P6 for RP(2); 
4.2: 2,3, Supp:1 
4.3: 1-6, Supp:1,5,6		Lab Exercise 8: Due 4/6
Draw a sketch for Desargue's theorem in the plane.
11 3/29 More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry. 

3/31 No Class.
CC Day.		 4/2 Proofs of some basic projective geometric facts.
Triangle Coincidences (Perpendicular Bisectors- the circumcenter)
M&I:4.1, 4.2, 4.3, 2.4
No Lab this week.
12 4/5 Duality Theorem and Desargues.  4/7Conic Sections.
Pascal and More Duality
4/9 Complete quadrangles Postulate 9. 
Projective transformations. Perspectivities and Projectivities. 
M&I: 
4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm)		 Due : 4/21
M&I:4.5:2; 4.6:7,8,9; 4.7:4,7 
4.10: 4,5,9,10
[Prove P9 for RP(2),optional] 		Lab Exercise 9: Due 4/13
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 4/12 Conics
Pascal's Theorem ?
More on coordinates and transformations. 		 4/14 Projectivities. Perspective  4/16Transformations of lines with homogeneous coordinates. 4.10, 5.4, 2.4 
4.11,
Lab Exercise 10: Due 4/20
A.1. Construct a sketch showing ABC on a line perspectively related to A'B'C' on a second line with center O.
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.
B.1'. Draw a dual sketch for the figure in problem 1.  2'. Draw a dual sketch for the figure in problem 2. 
C.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.
D 3'. Draw a dual sketch for the figure in problem 3 
14 4/19 Projectivities in 3 space: More on Projective Line Transformations with Coordinates. 
4/21 Harmonic sets: uniqueness and construction of coordinates for a Projective Line, Plane, Space.  4/23   
More on Transformations, Coordinates and Harmonic sets. .		 5.1,5.4
Due: 4/28
M&I: 4.10:1,3,6,7; 
5.1:5; 5.4:1-8,10;  5.5: 2,3,7		Lab Exercise 11: Due 4/27
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*. 
15 4/26
Matrices for familiar Planar Projective Transformations. 		 4/28  Conics revisited.
Inversion and the final exam.
 4/30 Quiz #3
Inversion.		 5.1,5.2, 5.3,5.5, 5.7, 6.1, 6.2
Lab Exercise 12: Due 5/4 Use five points and Pascal's Theorem to construct a conic.
16 5/3 Projective generation of conics
Pascal's and Brianchon's Theorems.		 5/5 Inversion angles, circles and lines.
 5/7
The Big picture in Summary.
Student Presentations		 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}.