TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo,
A.W. (1969) 
ON LINE
through Moodle.
The
Elements by Euclid, 3 volumes, edited by T.L. Heath, Dover
(1926)
Proof in Geometry by A.I Fetisov, Mir (1978)
Flatland By E. Abbott, Dover.
Week  Monday  Wednesday  Friday  Reading/Videos for the week.  Problems
Due on Wednesday of the next week 

1  No Class 
12109 1.1 Beginnings 
123 
M&I:1.1,
1.2
E:I Def'ns, etc. p1535; Prop. 112,22,23,47 A:.Complete in three weeks 
Due:13009 M&I p5:18,11 
2  126 The Pythagorean Theorem

More on Pythagoras and Euclid. 
130 Prop 2 Begin disssection theory, 
M&I
1.2, 1.3
E: I Prop. 16, 2732, 3545. 
Due: 24 M&I: p10:1,2,5,10,1113 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. [ HELP! Proof outline for the midpoint proposition.] 
3  2/2Convexity defined. Equidecomposable Polygons Constructions Isometries 1.1 Def'ns Objects  2/4 1.2 Constructions 1.4 Continuity  2/6 1.3 Geometry: Constructions and numbers  M&I
1.3,1.4
E: III Prop. 13, 1418, 20, 21, 10 Here’s looking at Euclid F. Sect. 11, 25, 31 Watch Equidecomposable Polygons  Due: 213 (changed 213!) M&I: p11: 1619, 24, *27 Problem Set 1 
4  2/9 More on continuity and rational points. 
2/11 Start Inversion Orthogonal Circles 
2/13More on Cantor Similar triangles Odds and ends. Coordinates. 
M&I:1.5,
1.6,
2.1
E: V def'ns 17;VI: prop 1&2 F. Sect. 32 
Due 218 M&I: p17: 5, 811 p23: 9,10 (analytic proofs) 
5  2/16 Odds and ends  2/18 Odds and ends 
2/20Start: Transformations  Isometries Transformations  Isometries. 
M&I:
2.1, 2.2
E: V def'ns 17;VI: prop 1&2 E:IV Prop. 35 Isometries (Video # 2576 in Library) . 
Due: 225 M&I:1.6:112,17,18 Problem Set 2 
6  2/23More Isometries: Coordinates and Transformations classification 
2/25 More on Isometries 
2/27 .... 
M&I: 2.1,2.2  Due 34 Problem Set 3 (Isos Tri) [3 Points for every distinct correct proof of any of these problems.] 
7 
3/2Finish Classification of Planar Isometries. Begin Affine Geometry 
3/4
Isometries and symmetries 
3/6
Proportion and Similarity 
M&I: 2.2 again;3.1,3.2, 3.5  Quiz #1 on 3/11 See Sample quiz on Moodle. 
8 
3/9 More on Similarity 
3/11 Inversion and Affine Geometry (planar coordinates) the Affine Line.  3/13 Constuctions with Straight edge and compass.  M&I:3.6,
3.4,3.7 View video (in Library #4376 ) on "Central similarities" from the Geometry Film Series. (10 minutes) View video (in Library #209 cass.2) on similarity (How big is too big? "scale and form") "On Size and Shape" from the For All Practical Purposes Series. (about 30 minutes) 

9 Spring break  3/16 No Class 
3/18 No Class  3/20 No Class  
10  3/23 Visualizing
the affine plane. Seeing the infinite.Affine geometry Homogeneous coordinates and visualizing
the affine plane. 
3/25 More on Homogeneous coordinates for the plane.A first look at a "Projective plane." Axioms, consistency, completeness and models. 
3/27Begin Synthetic Geometry [Finite] Algebraicprojective geometry: Points and lines. Spatial and Planar Axioms for 7 point geometry. Begin Synthetic Projective Geometry 
A noneuclidean universe. M&I :3.1, 3.2, 3.5, 3.6, 3.7; 4.1 
Due : 4/1 M&I: 3.5: 1,3,4,5,10,11 3.6: 3,715 3.7: 1,4,7,10,13 
11  3/30 Homogeneous Coordinates with Z_{2 }and Z_{3} More on Finite Synthetic Geometry and models. Proof of Desargues' Theorem 
4/1 Projective Geometry Visual/algebraic and Synthetic..Axioms 16 Projective Planes. .  4/3. More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry. 
M&I:4.1, 4.2, 4.3, 2.4  
12  4/6 Proofs of some basic projective geometric facts.  4/8
Triangle Coincidences (Perpendicular Bisectors the circumcenter) Applications of Projective Geometry Postulates.16 
4/10 Quiz #2 Desargue's Theorem and Duality 
Due 4/15 M&I: 4.1:7,15,16; Prove P6 for RP(2); 4.2: 2,3, Supp:1 4.3: 16, Supp:1,5,6 

13  4/13 Conic Sections.
More Duality 
4/15 More Postulates Sections (more duality) Complete quadrangles Postulate 9. 
4/17 Perspectivities and Projectivities. Projective transformations. Conics Pascal's Theorem ? More on coordinates and transformations. 
M&I:
4.5,4.6(p9497).4.7, p105108 (Desargues' Thrm) Watch video: "Conics" [VIDEO #628]. Watch video: For All Practical Purposes on conicsFAPP on conics[HSU Video #209 3. Standup conic (Conic sections) (ca. 30 min.)]. WatchVideo: Orthogonal Projection [Video #4223] 
Due : 4/22 Problem Set 4 
14  4/20Projectivities. Perspective  4/22Transformations of lines with homogeneous coordinates  4/24 Projectivities in 3 space: More on Projective Line Transformations with Coordinates. Begin Harmonic Sets and Construction of Coordinates.  4.10, 5.4, 2.4
4.11 
Due: 4/29 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] 
15 Final exam available 4/30. 
4/27Harmonic sets: uniqueness and construction of coordinates for a Projective
Line, Plane, Space. 
4/29More on Transformations, Coordinates and Harmonic sets.  5/1  5.1,5.4;,5.2, 5.3,5.5, 5.7, 6.1, 6.2  
16 Final projects are due for review Tuesday, May 5th. 
5/4
Projective generation of conics Pascal's and Brianchon's Theorem. 
5/6 Matrices for familiar Planar Projective Transformations. Inversion angles, circles and lines? 
5/8 The Big picture in Summary. 








Due: M&I: 4.10:1,3,6,7; 5.4:18,10; 5.5: 2,3,7 


Due: M&I: 4.10:1,3,6,7; 5.1:5; 5.4:18,10; 5.5: 2,3,7 

16 
Conics revisited. Inversion and the final exam. Quiz #3 

. Student Presentations 
A Project Fair will be organized for displays and presentations during the last day of class. Details will be discussed later.
Guidelines for Preliminary Proposals:
Tiling patterns  tesselation 3d tiling MC Escher perspective Curves: conics, etc. optical illusions knots fractals Origami Kaleidescope Symmetry The coloring problem Patterns in dance and other performance arts Flatland sequel (4d)  Maps Juggling structural Rigidity dimension Polyhedra bridgemaking (architecture) Models (3d puzzles) paper mache or clay mobiles sculpture A play  movie build three dimensional shapes power point performance website 
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.]
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 P_{sqrt(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} 
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''= T_{l}(B). Prove that if B' is not equal to B'' then A' lies on the perpendicular bisector of the line segment B''B'.
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.
1. Use an affine line with P_{0} , P_{1} , and P_{inf }given. Show a construction for P_{1/2} and P_{2/3}.
2. Use an affine line with P_{0} , P_{1} , and P_{inf }
given. Suppose x > 1.
Show a construction for Px^{2} and Px^{3} 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}.