Mathematics 133
Notes from November 18, 2020, lecture session

The recording for this lecture is available online as  
http://math.ucr.edu/~res/math133-2020/Nov18.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/week7/geometrynotes03c.f13.pdf   http://math.ucr.edu/~res/math133-2020/week7/geometrynotes03d.f13.pdf   The third  and fourth parts of the third chapter of the course notes.   The lecture surveyed the material in Sections III.7  and III.8.   However, the only part of Section III.7 that might be covered on quizzes or the final examination are the formulas for the altitudes of a triangle which follows from the proof of Heron's Theorem (Sections III.7.6), and in Section III.8 students are only resposible for knowing the statements about choosing rectangular coordinate systems and understanding how to use them in proofs of more simply stated geetrical results (such as basic locus theorems). 
https://math.ucr.edu/~res/math133-2020/week7/Whiteboard.nov18-1.png   Drawing of the interior of a convex pentagon; the reader should understand the concept of interior well enough to draw the interior of a convex quadrilateral.   A "good" area theory should be able to yield the areas of regions formed by the union of a convex polygon and its interior.  
https://math.ucr.edu/~res/math133-2020/week7/Whiteboard.nov18-2.png   Whiteboard deconstruction for the derivation of the area formula for a regular polygon. (The example here is supposed to be a regular hexagon, but due to Zoom limitations it does not really look like a regular hexagon.  Use your imagination ... )  
https://math.ucr.edu/~res/math133-2020/week7/Whiteboard.nov18-3.png   Application of Heron's Formula and the theory of similar triangles to prove an angle-angle-area congruence theorem.   The reason for mentioning this result will become apparent in Chapter V.  
https://math.ucr.edu/~res/math133-2020/week7/pythagorean-thm.pdf   A classic wordless proof of the Pythagorean Theorem using areas.  
https://math.ucr.edu/~res/math133-2020/week7/history05f.pdf   A generalization of the Pythagorean Theorem due to Pappus.  This proof is a refinement of the area-theoretic proof of the Pythagorean Theorem in Euclid's Elements.
https://math.ucr.edu/~res/math133-2020/week7/coordinatization-from-Moise.pdf   A textbook description of the introduction of rectangular coordinates on a synthetic Euclidean plane; one particularly important point is to see how much choice one has in setting up a coordinate system.  
https://math.ucr.edu/~res/math133-2020/week7/Whiteboard.nov18-4.png   Simplified description for the introduction of rectangular coordinates and the amount of choice one has in setting up a coordinate system.       
https://math.ucr.edu/~res/math133-2020/week7/history11b.pdf  Pages 2 and 3 only.  Comparison of a synthetic and (much shorter) analytic proof of a basic locus theorem. 
https://math.ucr.edu/~res/math133-2020/week7/locus-problems.pdf    https://math.ucr.edu/~res/math133-2020/week7/locus-problems1.pdf https://math.ucr.edu/~res/math133-2020/week7/locus-problems2.pdf https://math.ucr.edu/~res/math133-2020/week7/locus-problems3.pdf  Several files which discuss such problems and include several solved examples.
https://math.ucr.edu/~res/math133-2020/week7/locus-aristotle.pdf   Discussion of a locus theorem which appears in the writings of Aristotle; a special case of this statement is also a homework exercise (see Problem 2 in the last file of the previous list).  The file also contains an animated link illustrating this result.
https://math.ucr.edu/~res/math133-2020/week7/math133hw03.pdf   https://math.ucr.edu/~res/math133-2020/week7/math133hw03a.pdf   Lists 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.