MATH 371 Assignments Spring, 1998

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TEXTS: Fundamentals of Geometry by B.Meserve and J. Izzo, A.W. (1969)
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.

Week Monday Wednesday Reading Problems 
Due on Wednesday 
of the next week
1 M.L.King Day 1/21 
1.1 Beginnings 
What is Geometry? 
The Pythagorean Theorem 
Intro to Geometer's Sketchpad
E:I Def'ns, etc. p153-5; 
Prop. 1-12,22,23,47 
A:.Complete in three weeks
M&I p5:1-8,11
2 1/26 
1.1 Def'ns- Objects 
1.2 Constructions 
Equidecomposable Polygons
1.2 More on Constructions 
M&I: 1.2 
E: I Prop. 16, 27-32, 35-45.
M&I: p10:1,2,5,10,11-13
3 2/2 
1.3 Geometry and numbers
2/4 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 
Problem  Set 1
4 2/9 2/11 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
5 2/16 Coordinates and Transformations 
Begin Affine Geometry
2/18 Affine Geometry M&I: 2.1, 2.2, 
E:IV Prop. 3-5
M&I: p23:9,10 
Problem Set 2
6 2/23 More affine geometry. 2/25 More Affine Geometry (planar coordinates) M&I: 2.1,2.2
7 3/2 Homogeneous Coordinates 
Begin Synthetic Geometry 
Finite Geometry
3/4 Continuation on finite geometries and coordinates. M&I: 3.1,3.2, 3.5  M&I:3.5: 1,3,4,5,10,11 
3.6: 3,7-15 
3.7: 1,4,7,10,13 
Problem Set 3 (Isos Tri)
8 3/9 Homogeneous Coordinates with Z2 and Z3 
Begin algebraic -projective geometry.
3/11 Begin Synthetic Projective Geometry 
(Quiz #1) 
M&I:3.6, 3.4,3.7 Problem Set 4
9 Spring break 3/16 No Class 3/18 No Class
10 3/23 Synthetic Projective Geometry -Planes 
Triangle Coincidences 
3/25 Duality Planar 
Desargues' Theorem 
Inversion properties.
M&I:4.1, 4.2, 4.3, 2.4 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
11 3/30 Sections 
Transformations of lines with homogeneous coordinates.
4/1Complete quadrangles 
Projective transformations 
M&I:4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm) 
4.10, 5.4
M&I:4.5:2; 4.6:7,8,9; 4.7:4,7 
Prove P9 for RP(2)
12 4/6 Projectivities 
Pascal's Theorem 
More on coordinates and transformations.
4/8 (Quiz# 2)
Begin Harmonic sets 
4.11,5.1,5.4 M&I: 4.10:1,3,5,6,7; 
5.1:5; 5.4:1-8
13 4/13 4/15
14 4/20 4/22
15 4/27 4/29
16 5/4 5/6
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 P x 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}.