- Course Description
- Some www sites related to geometry and visual mathematics.
- Details/summaries of the actual lectures.

Back to Martin Flashman's Home Page :) Last updated: 3/17/98

TEXTS: *Fundamentals of Geometry* by B.**M**eserve and J.
**I**zzo, A.W. (1969)

*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 | Reading | Problems Due on Wednesday of the next week |
---|---|---|---|---|

1 | M.L.King Day | 1/21 1.1 Beginnings What is Geometry? The Pythagorean Theorem Intro to Geometer's Sketchpad |
M&I:1.1 E:I Def'ns, etc. p153-5; Prop. 1-12,22,23,47 A:. Complete in three weeks |
M&I p5:1-8,11 |

2 | 1/26 1.1 Def'ns- Objects 1.2 Constructions Transformations, Equidecomposable Polygons |
1/28 1.2 More on Constructions Isometries |
M&I: 1.2 E: I Prop. 16, 27-32, 35-45. |
M&I: p10:1,2,5,10,11-13 |

3 | 2/2 1.3 Geometry and numbers |
2/4 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 Problem Set 1 |

4 | 2/9 | 2/11 | 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 |

5 | 2/16 Coordinates and Transformations Inversion Begin Affine Geometry |
2/18 Affine Geometry | M&I: 2.1, 2.2, E:IV Prop. 3-5 |
M&I: p23:9,10 Problem Set 2 |

6 | 2/23 More affine geometry. | 2/25 More Affine Geometry (planar coordinates) | M&I: 2.1,2.2 | |

7 | 3/2 Homogeneous Coordinates Begin Synthetic Geometry Finite Geometry |
3/4 Continuation on finite geometries and coordinates. | M&I: 3.1,3.2, 3.5 | M&I:3.5: 1,3,4,5,10,11 3.6: 3,7-15 3.7: 1,4,7,10,13 Problem Set 3 (Isos Tri) |

8 | 3/9 Homogeneous Coordinates with Zand
_{2 }Z _{3}Begin algebraic -projective geometry. |
3/11 Begin Synthetic Projective Geometry (Quiz #1) |
M&I:3.6, 3.4,3.7 | Problem Set 4 |

9 Spring break | 3/16 No Class | 3/18 No Class | ||

10 | 3/23 Synthetic Projective Geometry -Planes Triangle Coincidences |
3/25 Duality Planar Desargues' Theorem Inversion properties. |
M&I:4.1, 4.2, 4.3, 2.4 | 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 | 3/30 Sections Perspective Transformations of lines with homogeneous coordinates. |
4/1Complete quadrangles Projective transformations |
M&I:4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm) 4.10, 5.4 |
M&I:4.5:2; 4.6:7,8,9; 4.7:4,7 Prove P9 for RP(2), 4.10:4,5,9,10 |

12 | 4/6 Projectivities Conics Pascal's Theorem More on coordinates and transformations. |
4/8 (Quiz# 2) Begin Harmonic sets Breath |
4.11,5.1,5.4 | M&I: 4.10:1,3,5,6,7; 5.1:5; 5.4:1-8 |

13 | 4/13 | 4/15 | ||

14 | 4/20 | 4/22 | ||

15 | 4/27 | 4/29 | ||

16 | 5/4 | 5/6 |

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 } |

4. **Construct with SEC** on a Euclidean line sqrt(
sqrt(5)/sqrt(3) + sqrt(6) ).

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 P x
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}.