TEXTS: Fundamentals of Geometry by B. Meserve and J.
Izzo,
A.W. (1969) 
ON LINE at 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  Tuesday  ??  Thursday  Reading/Videos for the week.  Problems Due on Wednesday of the next week 

1  1/18 1.1
Beginnings What is Geometry? 

1/20 
M&I:1.1,
1.2 E:I Def'ns, etc. p1535; Prop. 112,22,23,47 A:.Complete in three weeks 
Due:1/26 M&I p5:18,11 
2  1/25
Euclid. Prop 13 The Pythagorean Theorem 
. The Pythagorean Theorem 1.2 
1/27 More Euclid Constructions. 
M&I
1.2,
1.3 E: I Prop. 16, 2732, 3545 Watch online: Here’s looking at Euclid. 
Due: 2/2 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 Reading Report due: 2/1  2/1
Equidecomposable Polygons Constructions Isometries 1.1 Def'ns Objects 1.2 Constructions 1.3 Geometry: Constructions and numbers 
1.4 Continuity  2/3
Breather: Start: Transformations  Isometries 
M&I
1.3,1.4 E: III Prop. 13, 1418, 20, 21, 10 F. Sect. 11, 25, 31 Watch Equidecomposable Polygons 
Due: 2/9 M&I: p17:5, 811 p11: 1619, 24, *27 Problem Set 1 
4Preliminary Project Proposals first review 5 p.m. Wednesday, February 9th.  2/8 More on continuity and rational points. Similar triangles  2/10 More on similar triangles and rational
points. Start Inversion. Orthogonal Circles 
M&I:1.5,
1.6,
2.1
OPTIONAL[E: V def'ns 17;VI: prop 1&2 ] F. [Sect. 32] pp 5561on Axioms of Continuity 

5 Reading Report due: 2/15  2/15
Coordinates and Proofs Inversions and orthogonal circles. Odds and ends on continuity axiom More on Cantor 
Isometries /  2/17
Transformations  Isometries. coordinates/ 
M&I:
2.1,
2.2 OPTIONAL [E: V def'ns 17;VI: prop 1&2 ] E:IV Prop. 35 Isometries (Video # 2576 in Library) . 
Due : 2/23 [Previously due 2/16] M&I: p23: 9,10 (analytic proofs) Problem Set 2 
6. 
2/22 More Isometries: classification 
More
on
Isometries 
2/24 .... Finish Classification of Planar Isometries.  M&I: 2.1,2.2  
7 Reading Report due: 3/1 
3/1
Isometries
and
symmetries 
More
on
Similarity 
3/Begin
Affine
Geometry Proportion and Similarity 
M&I: 2.2 again;3.1,3.2, 3.5  
8Quiz #1 on Thursday 3/10 in class  3/8
3
Inversion
and
Affine
Geometry
(planar
coordinates) The Affine Line. 
Affine geometry Homogeneous coordinates and visualizing the affine plane.  3/10
Visualizing
the
affine
plane.
Seeing
the infinite. More on Homogeneous coordinates for the plane. 
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) 
Due 3/23 M&I:1.6:112,17,18 Problem Set 3 (Isos Tri) [3 Points for every distinct correct proof of any of these problems.] 
9 Spring break  3/15 No Class  No Class  3/17 No Class  
10
Reading Report due: 3/24 Project progress report Thursday, March 25th 
3/22
More
on
The
affine
Plane
 A first look at
a "Projective plane." Axioms, consistency, completeness and models. A noneuclidean universe. 
Begin
Synthetic
Geometry
[Finite]
Algebraicprojective
geometry:
Points
and lines. Spatial and Planar 
3/24Axioms
for
7
point
geometry. Begin Synthetic Projective Geometry 
M&I
:3.1,
3.2,
3.5,
3.6,
3.7;
4.1 
Due
:
3/30 M&I: 3.5: 1,3,4,5,10,11 3.6: 3,715 3.7: 1,4,7,10,13 Problem Set 4 
11 
3/29
Homogeneous
Coordinates
with
Z_{2 }and Z_{3} More on Finite Synthetic Geometry and models. Proof of Desargues' Theorem Projective Geometry Visual/algebraic and Synthetic.. 

3/31
No
Class
CC Day 
M&I:4.1, 4.2, 4.3, 2.4  
12 Reading Report due: 4/7 (changed 43)  4/5
Axioms
16 Projective Planes. More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry. Proofs of some basic projective geometric facts. 
Applications
of
Projective
Geometry
Postulates.16 
4/7
Triangle
Coincidences
(Perpendicular Bisectors the circumcenter) Desargue's Theorem and Duality 
M&I:4.1, 4.2, 4.3, 2.4  Due: 4/13 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 Problem Set 5 
13 quiz #2 4/14 
4/12Conic
Sections.
Pascal and More Duality 
Projective transformations. Perspectivities and Projectivities. 
4/14Sections 
M&I:
4.5,4.6(p9497).4.7, p105108 (Desargues' Thrm) 

14 Reading Report due: 4/19  4/19
Perspective
More duality. Graphs and duality. Complete quadrangles Postulate 9. 
4/21Projectivities.
Transformations
of
lines
with
homogeneous
coordinates. Conics Pascal's Theorem ? More on Projective Line Transformations with Coordinates. Begin Harmonic Sets and Construction of Coordinates. 
4.10,
5.4,
2.4
4.11 
Due
:4/27 M&I:4.5:2; 4.6:7,8,9; 4.7:4,7 4.10: 4,5,7,9,10 [Prove P9 for RP(2),optional] 

15 Projects are due 5pm Thursday, 4/29 
4/26More
on
coordinates and transformations. Projectivities
in
3
space: Harmonic
sets:
uniqueness
and
construction
of coordinates for a Projective
Line, Plane, Space. Projective generation of conics 
More on Transformations, Coordinates and Harmonic sets. .  4/28Matrices
for
familiar
Planar
Projective
Transformations.

5.1,5.4;,5.2, 5.3,5.5, 5.7, 6.1, 6.2  Due: OPTIONAL! M&I: 4.10:1,3,6; 5.1:5; 5.4:18,10; 5.5: 2,3,7 
16 Reading Report due: 5/3  5/3Conics
revisited.
Inversion and the final exam. Quiz #3 
Inversion angles, circles and lines.  5/5
Pascal's and Brianchon's Theorem. The Big picture in Summary. . 


17 Final Exam Week. 
Office Hours for Exam Week MTWRF 8:1510:00 and by appointment or chance 
Student Presentations will be made Tuesday 510 3:004:50  Final Exam DUE Friday, May 13, before 5 P.M. 
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.
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.
1. 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.
2. 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}.