Agenda

Quiz

Old Business

1. Return homework; discuss homework policies.
2. Respond to questions/problems relating to the homework assignment due Wednesday.
3. Summary of main themes so far introduced and the specific expections for students.
   1. The systematization of knowledge. Natural and formal systems.
   2. The deductive method; primitive terms, defined terms, postulates, propositions, proofs.
   3. Congruence of segments and angles.
      • Primitive notion.
      • Reflexive, symmetric and transitive relation.
      • The radii of a circle are all congruent to one another.
      • Any two right angles are congruent.
   4. Congruence of triangles defined. SAS, SSS
   5. Contructing perpandicular bisectors and angle bisectors.
   6. Euclid Book I. The first part (Propositions 1--26) does not use the Parallel Postulate.
   • Propositions 1--3. Copying a segment whereever you please. (A collapsing compass can do everything a compass with a memory can do.
   • Propositions 4--8. Basic properties of Triangles.
     • Prop. 4 is SAS. The proof uses "superposition", and therefore involve ideas not stated explicitlt in the Postulates. Modern systems take SAS as a postulate.
     • Prop. 5, Prop 6. A triangel is isoceles IFF two angles are equal.
     • prop. 8. SSS.
   • Propositions 9--12. Construction of angle bisectors and various perpendiculars.
   • Propositions 13--15. Facts about angle supplements and vertical angles.
   • Propositions 16--21. Various inequalities involving angles and sides of a traingle.
   • Propositions 22--23. Constructions: To copy a triangle. To copy an angle.
   • Propositions 24--26. Comparing two traingles in which two sides are congreunt to two sides respectively. This leads to ASA and AAS in Prop. 26.
   7. Euclid Book I. The second part (Propositions 27--48) uses the Parallel Postulate.

New Business: Similarity of figures.

Definition. In saying that ABC...D is a configuration of points, I mean that ABC...D is a collection of at least three--possibly more---points, labelled A, B, C, ..., etc.

Definition. Suppose ABC...D and A'B'C'...D' are configurations of points. I say that ABC...D is similar to A'B'C'...D' if:

• The ratios |AB|:|A'B'|, |AC|:|A'C'|, ...etc. are all the same (i.e., for any points X and Y in ABC...D and corresponding points X' and Y' in A'B'C'...D', the ratio |XY|:|X'Y'| is the same.
• For any three points X, Y and Z in in ABC...D and corresponding points X', Y' and X' in A'B'C'...D', the angle XYZ and angle X'Y'Z' are congruent.

Remark. In the second definition, the term "ratio" appears. We have not yet defined this. There are two approaches. 1) If we are working is a geometric system in which segments have numerical length, then the ratio |AB|:|A'B'| means the number obtained by dividing the length of |AB| by the length of |A'B'|. 2) If, as in ancient Greek mathematics, segments are not assumed to have a numerical length, then the ratio must be defined differently. To see what Euclid does, consult Book V and Joyce's comments on Book V.

There are numerous definitions of similarity floatimg around. Typical of what you may find on the web is the following:

Similar figures have the same shape (but not necessarily the same size) and the following properties:

• Corresponding sides are proportional. That is, the ratios of the corresponding sides are equal.
• Corresponding angles are equal.

(Source: http://www.mathsteacher.com.au/year10/ch07_basic_geometry/05_similar/figures.htm

For next time

1. "Thales Theorem" (Euclid as Proposition 2 of Book VI) and its converse. Warning: Propositions about similarity rest upon propositions concerning parallelism from the second part of Book I. In our initial study of similarity we will try to identify these propositions, but we will not study their proof till later.
2. The Sea Island Problem.
3. Theorems of Pappus and Desargues.

Homework due Wednesday, February 1.

1. Problems 1--5 on the January 25 handout (available for download here) .

2. Read about the Sea Island Problem here.