M ATH 532/736I, L ECTURE N OTES 10
Notes on Translations and Rotations
x y which we will sometimes write as We associate with each point (x, y) the column 1 T (x, y, 1) . A translation of the Euclidean plane is a function f which maps each point (x, y) t
M ATH 532/736I, L ECTURE N OTES 9
The Nine-Point Circle Theorem. Let A, B and C be the three vertices of a triangle. Let MA be
the midpoint of BC, MB be the midpoint of AC, and MC be the midpoint of AB. Let APA be
an altitude for ABC (so PA is on BC). Let
E XAMPLES ON T RANSLATIONS AND ROTATIONS
(Lecture Notes for Math 532, taught by Michael Filaseta)
1. Let ABC be given, and let MB denote the midpoint of side AC and let MC - - denote the midpoint of side AB. Show that MB MC is parallel to BC and that the
M ATH 532/736I, L ECTURE N OTES 8
Theorem 1. Let A and B be distinct points. Then C is on AB if and only if there is a real number
t such that C = (1 t)A + tB.
Theorem 3. If A, B and C are points and there exist real numbers x, y, and z not all 0 such tha
M ATH 532/736I, L ECTURE 3
1. Finish Previous Notes 2. Homework: Assign further problems from Homework 1. Quiz: 01/27/09, Tuesday 3. Finite Projective Planes (Given a positive integer n called the order.) Axiom P1: There exist at least 4 points no 3 of wh
M ATH 532/736I, L ECTURE 2
1. Finish Previous Notes 2. Assignment: Problems 1, 2, 8, 9 and 10 from Homework 1. 3. Components of an Axiomatic Systems: (i) Undened Terms (points, lines) (ii) Dened Terms (parallel) (iii) Axioms (iv) A system of logic (if A o
Axioms for a Finite Projective Plane of Order n
Axiom P1. There exist at least 4 distinct points no 3 of which are collinear. Axiom P2. There exists at least 1 line with exactly n + 1 points on it. Axiom P3. Given any 2 distinct points, there exists exact
M ATH 532/736I, L ECTURE 1
1. Hand out and go over syllabus. 2. Class photos. 3. No homework (today). 4. Logic in the Last Century: Are there statements that can be made in mathematics which are true but which we cannot prove? Remark 1: The answer cannot
M ATH 532, 736I: M ODERN G EOMETRY
Name Practice Test #1 (1) State the axioms for a finite affine plane of order n. (Number or name the axioms so you can refer to them.)
(2) Two points have been circled in the 7 7 array of points below. Using the model fo
A T HEOREM C ONCERNING A FFINE P LANES
Theorem: In an afne plane of order n, each point has exactly n + 1 lines passing through it.
Lemma. If is a line with exactly n points on it (in a nite afne plane of order n) and A is a point
not on , then there are
M ATH 532, 736I: M ODERN G EOMETRY
Test 1 Solutions
Test 1 (1992):
Part I:
(1) Axiom P1:
Axiom P2:
Axiom P3:
Axiom P4:
There exist at least 4 distinct points no 3 of which are collinear.
There exists at least 1 line with exactly n + 1 (distinct) points on
M ATH 532, 736I: M ODERN G EOMETRY
Some Solutions to Old Final Exam Problems
Final Exam (1992):
Part II:
(2) (a) Let a, b, x, and y be such that B = (a, b) and A = (x, y ). Then
TB R,A = T(a,b) R,(x,y)
10a
cos() sin() x(1 cos() + y sin()
= 0 1 b sin() cos
Math 532: Quiz 2
Name
Axiom P1: There exist at least 4 points no 3 of which are collinear.
Axiom P2: There exists at least 1 line with exactly n + 1 (distinct) points on it.
Axiom P3: Given 2 distinct points, there is exactly 1 line that they both lie on.
Math 532: Quiz 2
ANSWERS
Name
Axiom P1: There exist at least 4 points no 3 of which are collinear.
Axiom P2: There exists at least 1 line with exactly n + 1 (distinct) points on it.
Axiom P3: Given 2 distinct points, there is exactly 1 line that they both