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.
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 
M&I:1.1,
1.2
E:I Def'ns, etc. p1535; Prop. 112,22,23,47 A:.Complete in three weeks 
Due: 1/28 M&I p5:18,11  
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, 2732, 3545. 
Due: 2/4
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. 
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. 13, 1418, 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, 811 p11: 1619, 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 17;VI: prop 1&2 F. Sect. 32 
Due: 2/11
Changed M&I: p17:5, 811 p11: 1619, 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. 35 
Due 2/25: M&I: p23: 9,10 (analytic proofs) M&I:1.6:112,17,18 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 Z_{2 }and Z_{3} More on Finite Synthetic Geometry and models. 
M&I:3.6, 3.4,3.7  Due: 3/24 Problem Set 4 3.6: 3,715 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 noneuclidean universe. 
3/24 Algebraicprojective 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 16 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: 16, 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(p9497).4.7, p105108 (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; 5.4:18,10; 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 
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}.