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Back to Martin Flashman's Home Page :) Last updated: 1/27/03

TEXTS: *Fundamentals of Geometry* by B. **M**eserve and J. **I**zzo,
A.W. (1969) -
ON LINE with HSU ONCORE
*The
Elements *by **E**uclid, 3 volumes, edited by T.L. Heath, Dover
(1926)
*Proof in Geometry* by A.I **F**etisov, Mir (1978)
*Here's Looking at Euclid...*, by J.Petit, Kaufmann (1985).
*Flatland *By E. **A**bbott, Dover.

Week | Monday | Wednesday | Friday |
Reading for the week. | Problems
Due on Wednesday of the next week |
---|---|---|---|---|---|

1 | 1/20 No Class | 1/23 1.1 Beginnings
What is Geometry? The Pythagorean Theorem |
1/25Intro to Geometer's Sketchpad/ Wingeom
Transformations |
M&I:1.1,
1.2
E:I Def'ns, etc. p153-5;
Prop. 1-12,22,23,47 A:. Complete in three weeks |
M&I p5:1-8,11 Due: 1/29 |

2 | 1/27 The Pythagorean Theorem | 1/29
1.2 Equidecomposable Polygons |
1/31 | M&I
1.2, 1.3
E: I Prop. 16, 27-32, 35-45. |
M&I:
p10:1,2,5,10,11-13 Due:2/6
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: 1/31
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 | 2/3 More on Constructions
Isometries 1.1 Def'ns- Objects 1.2 Constructions |
2/5 1.3 Geometry and numbers | 2/7 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 Due:2/13 Problem Set 1 Due:2/13 Lab Exercises 2: Due by 2/8.
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 parallelogram. |

4 | 2/10 Transformations - Isometries | 2/12 Coordinates and Transformations | 2/14 Inversion
Begin Affine Geometry |
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
Due 2/20
Problem Set 2 Due 2/20 Lab Exercises 3:
Due 2/14. 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 | 2/17 Affine Geometry | 2/19 Affine Geometry | 2/21 More affine geometry. | M&I:
2.1, 2.2,
E:IV Prop. 3-5 |
M&I:
p23: 9,10 (analytic proofs) Due 2/27
Lab Exercises 4:
Due 2/21.1. 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 | 2/24 Orthogonal circles and Inversion. | 2/26 More Inversion and Affine Geometry (planar coordinates) | 2/28 Seeing the infinite | M&I: 2.1,2.2 | Problem Set 3 (Isos Tri) Due
3/6 [4 Points for every distinct correct proof of any
of these problems.]
Lab Exercises 5:
Due 2/28.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 | 3/3 Homogeneous Coordinates. | 3/5More on Affine geometry- Homogeneous coordinates and visiaulizing the affine plane. | 3/7 More on Homoogeneous coordinates for the plane. | M&I: 3.1,3.2, 3.5 | Due 3/13 M&I:
3.5: 1,3,4,5,10,11
Lab Exercise 6: Due 3/7 . See Notes [Abridged]
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 | 3/10 Begin Synthetic Geometry [Finite]
Homogeneous Coordinates with Zand _{2 }Z_{3} |
3/12 More on Finite Synthetic Geometry and models. | 3/14 Algebraic-projective geometry: Points and lines. | M&I:3.6, 3.4,3.7 | Lab Exercise 7: Due 3/14 . 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).
Due: 3/27
Problem Set 4 3.6: 3,7-15 3.7: 1,4,7,10,13 |

9 Spring break | 3/17 No Class | 3/19 No Class | 3/21 | ||

10 | 3/24 Spatial and Planar Desargues' Theorem
Begin Synthetic Projective Geometry |
3/26 Projective Geometry -Visual/algebraic and Synthetic... Axioms 1-6 | 3/28 -Projective Planes.
Triangle Coincidences (Perpendicular Bisectors- the circumcenter) |
M&I:4.1, 4.2, 4.3, 2.4 | Lab Exercise 8: Due 3/28Draw a sketch for Desargue's theorem in the plane.Due 4/10. (changed 4-2)
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 | 3/31No Class.
CC Day. |
4/2 More on the axioms of Projective Geometry.
Proof of Desargues' Theorem |
4/4
Duality Theorem and Desargues. |
M&I:4.1, 4.2, 4.3, 2.4 | No Lab this week. |

12 | 4/7 Conic Sections.
Pascal and More Duality |
4/9 Complete quadrangles Postulate 9.
Projective transformations. |
4/11 Perspectivities and Projectivities.
Conics ? Pascal's Theorem ? |
M&I:
4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm) |
Lab Exercise 9: Due 4/11Pascal'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 | 4/14 More on coordinates and transformations. | 4/16 Projectivities. Perspective | 4/18Transformations of lines with homogeneous coordinates. | 4.10, 5.4, 2.4
4.11, |
Lab Exercise 10: Due 4/18
1. Construct a sketch showing ABC on a line perspectively related to A'B'C' on a second line with center O. 1'. Draw a dual sketch for the figure in problem 1. 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. 2'. Draw a dual sketch for the figure in problem 2. 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. 2'. Draw a dual sketch for the figure in problem 3 Due : 4/17
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] |

14 | 4/21 Projectivities in 3 space: More on Projective Line Transformations with Coordinates. Begin Harmonic sets | 4/23 Harmonics: uniqueness and construction of coordinates for a Projective Line, Plane, Space. | 4/25 A Non-Euclidean Universe. | 5.1,5.4 | Lab Exercise 11: Due May 2
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*.
3.(new) Use five points and Pascal's Theorem
to constuct a conic.
Due :5/1 M&I:
4.10:1,3,6,7;
5.1:5; 5.4:1-8,10; 5.5: 2,3,7 |

15 | 4/28 Projective genration of conics. More on Transformations, Coordinates and Harmonic sets. Matrices for familiar Planar Projective Transformations. |
4/30 | 5/2 Conics revisited. Pascal's and Brianchon's Theorems. Equations for conics. | 5.1,5.2, 5.3,5.5, 5.7, 6.1, 6.2 | |

16 | 5/5 The Big picture in Summary.
Inversion properties? |
5/7 Student Presentations | 5/9 | 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

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}.