TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo,
A.W. (1969) 
ON LINE with HSU ONCORE
through Moodle.
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  Friday  Reading/Videos for the week.  Problems
Due on Wednesday of the next week  Math 480 Lab Assignments 

1  1/15 No Class 
1/17 1.1 Beginnings

1/19 
M&I:1.1,
1.2
E:I Def'ns, etc. p1535; Prop. 112,22,23,47 A:.Complete in three weeks 
Due:1/24 M&I p5:18,11  
2  1/22 The Pythagorean Theorem
Lab: Intro to Geometer's Sketchpad/ Wingeom 
1/24 Convexity defined. The Pythagorean Theorem 1.2 
1/26 Guest Lecture on Euclid Here’s looking at Euclid 
M&I
1.2, 1.3
E: I Prop. 16, 2732, 3545. 
Due:
1/31 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.] 
Lab Exercises 1: Due: 1/29 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  1/29 Equidecomposable Polygons Constructions
Isometries 1.1 Def'ns Objects 1.2 Constructions 1.3 Geometry: Constructions and numbers  1/31 1.4 Continuity  2/2 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: 2/7 [extended to2/14] M&I: p17:5, 811 p11: 1619, 24, *27 Problem Set 1  Lab Exercises 2: Due by : 2/5 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  2/5 More on continuity and rational points. Similar triangles  2/7 More on Cantor Start Inversion. 
2/9Orthogonal Circles Odds and ends. Transformations  Isometries. Coordinates. 
M&I:1.5,
1.6,
2.1
E: V def'ns 17;VI: prop 1&2 F. Sect. 32 
Due 2/19 [Changed as of 29] M&I: p23: 9,10 (analytic proofs) M&I:1.6:112,17,18 Problem Set 2 
Lab Exercises 3: Due 2/12 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  2/12 More Isometries: Coordinates and Transformations  2/14 Isometries / 
2/16 coordinates/ 
M&I:
2.1, 2.2
E: V def'ns 17;VI: prop 1&2 E:IV Prop. 35 Isometries (Video # 2576 in Library) . 
Lab Exercises 4: Due 2/19 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  2/19 classification  2/21 More on Isometries 
2/23 .... Finish Classification of Planar Isometries.  M&I: 2.1,2.2  Due 2/28 Extended to 3/7! Problem Set 3 (Isos Tri) [1 Point for every distinct correct proof of any of these problems.] 
Lab Exercises 5: Due 2/26.
1. 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. 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. 
7 Quiz #1 on Monday in class. 
2/26Isometries and symmetries Begin Affine Geometry Proportion and Similarity 
2/28 More on Similarity 
3/2 Inversion and Affine Geometry (planar coordinates) the Affine Line.  M&I: 2.2 again;3.1,3.2, 3.5  Lab Exercise 6:
Due 3/5 1.Draw sketches for each of the following triangle coincidences: 1. Medians. 2. Angle Bisectors. 3. Altitudes. 4. Perpendicular Bisectors 

8 
3/5 Visualizing the affine plane. Seeing the infinite.  3/7 Affine geometry Homogeneous coordinates and visualizing the affine plane.  3/9 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) 
Lab Exercise 7: Due 3/19.
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  3/12 No Class  3/14 No Class  3/16 No Class  
10 
3/19 More on The affine Plane  A first look at a "Projective plane." Axioms, consistency, completeness and models. A noneuclidean universe. 
3/21 Begin Synthetic Geometry [Finite] Algebraicprojective geometry: Points and lines. Spatial and Planar 
3/23Axioms for 7 point geometry. Begin Synthetic Projective Geometry 
M&I :3.1, 3.2, 3.5, 3.6, 3.7; 4.1 
Due : 3/28 M&I: 3.5: 1,3,4,5,10,11 3.6: 3,715 3.7: 1,4,7,10,13 Problem Set 4 
Lab Exercise 8: Due 3/26 Draw a sketch for Desargues' theorem in the plane. Optional: Draw a spatial skectch for Desargues' Theorem 
11 
3/26 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..Axioms 16 Projective Planes. . 
3/28 More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry.

3/30 No Class.
CC Day. 
M&I:4.1, 4.2, 4.3, 2.4  Due 4/11!  watch for additional problems. 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 
Lab Exercise 9: Due 4/2 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. 
12 
4/2Proofs of some basic projective geometric facts. Triangle Coincidences (Perpendicular Bisectors the circumcenter) 
4/4 Applications of Projective Geometry Postulates.16 
4/6 Desargue's Theorem and Duality  No Lab this week.  
13 
4/9Conic Sections.
Pascal and More Duality 
4/11 Complete quadrangles Postulate 9.
Projective transformations. Perspectivities and Projectivities. 
4/13 Conics
Pascal's Theorem ? More on coordinates and transformations. 
M&I:
4.5,4.6(p9497).4.7, p105108 (Desargues' Thrm) 
Due : 4/18 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 10: Due 4/16
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. 
14 
4/16Projectivities. Perspective 
4/18 Transformations of lines with homogeneous coordinates.  4/20 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 


15 
4/23Harmonic sets: uniqueness and construction of coordinates for a Projective
Line, Plane, Space. Projective generation of conics 
4/25 More on Transformations, Coordinates and Harmonic sets. .  4/27Matrices for familiar Planar Projective Transformations. 
5.1,5.4;,5.2, 5.3,5.5, 5.7, 6.1, 6.2  Due: 5/2 M&I:
4.10:1,3,6,7;
5.1:5; 5.4:18,10; 5.5: 2,3,7 
Lab Exercise 11: Due 4/30 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*. 
16 
4/30Conics revisited. Inversion and the final exam. Pascal's and Brianchon's Theorem. Quiz #3 
5/2 5/5 Inversion angles, circles and lines. 
5/4The Big picture in Summary. . 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.
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}.