Mathematics 133
Notes from November 9, 2020, lecture session

The recording for this lecture is available online as  
http://math.ucr.edu/~res/math133-2020/Nov09.mp4  .

 Relevant directory files:
Here is a listing with comments on their significance.  These links are all clickable.
http://math.ucr.edu/~res/math133-2020/week6/geometrynotes03b.f13.pdf   The second  part of the third chapter of the course notes.   The lecture covered the remainder of Section III.4  and all the material in Section III.5.   The material on external angle bisectors at the end of Section III.4 will not be covered on quizzes or the final examination.
https://math.ucr.edu/~res/math133-2020/week6/thm3.4.6.pdf  Clarification of a crucial point in the proof; namely, for every point in the interior of angle ABC which arises in the proof, the foot of the perpendicular from that point to lines AB and BC meets these lines at points of the open rays (BA and (BC.
https://math.ucr.edu/~res/math133-2020/week6/similarity-classical.pdf   A summary of the steps needed to establish the basic similarity theorem using classical Greek synthetic methods.  The proof in the commensurable case shouild be understood actively, and the proof in the incommensurable casee should be understood passively.
https://math.ucr.edu/~res/math133-2020/week6/history03a.pdf   Proof of the Condition of Eudoxus, which gives criteria for two ratios of positive real numbers to be equal.  This is included for referennce purposes and will not be needed for quizzes and examinations.
https://math.ucr.edu/~res/math133-2020/week6/Whiteboard.nov09-2.png   Whiteboard summary for an example in the commensurable case.  The theorem cited is the Notebook Paper Theorem from the preceding document.
https://math.ucr.edu/~res/math133-2020/week6/metgeom.pdf   For this lecture, the most important parts are those involving similarity.  These consist of the subheading Similarities,  the proposition on pages 9 and 10, and the material on the last two pages.  
https://math.ucr.edu/~res/math133-2020/week6/Whiteboard.nov09-3.png   Reminder that the inner product of two n – dimensional column vectors v and w  (n x 1  matrices) is equal to the matrix product (Tv)w, where Tv denotes the matrix transpose of v.  The value may be viewed as a scalar because this is a product of a 1 x n row matrix (the transposed matrix) with an n x 1 column matrix.
https://math.ucr.edu/~res/math133-2020/week6/Whiteboard.nov09-4.png   Whiteboard discussion and visual example for a simple but fundamental realization theorem:  Given a subset S of the plane and a positive constant k, one can always find another subset T such that S is similar to T with ratio of similitude equal to k.  
https://math.ucr.edu/~res/math133-2020/week6/similarity.pdf   A summary of the discsussion of similarity in the preceding six documents.
https://math.ucr.edu/~res/math133-2020/week6/history06e.pdf   A generalization of the Pythagorean Theorem due to Thabit ibn Qurra.  There are two main types of approaches to proving the Pythagorean Theorem:   In Euclid's Elements this result is established using areas, but it is also possible to prove the result using similar triangles.  The proof for the result in this document is based upon the second approach.
https://math.ucr.edu/~res/math133-2020/week6/Whiteboard.nov09-5.png   Review of a standard identity for proportions which is used in the derivation of the Angle Bisector Theorem.  
https://math.ucr.edu/~res/math133-2020/week3/math133hw03.pdf   List of the strongly recommended homework assignments for Chapter III.  This listing includes assignments for all lectures from October 28 lecture through November 18.   Problems at roughly comparable levels of difficulty can be expected on quizzes and examinations.