©2019 An Li. All rights reserved.1MATH 228 Fall 2019Professor Thomas FoxContentsLecture 1–Wednesday, September 4, 2019..................................................................4History.........................................................................................................................4Lecture 2–Friday, September 6, 2019...........................................................................5Lecture 3–Monday, September 9, 2019........................................................................7The “converse” of a theorem.......................................................................................7Converse of Thales I................................................................................................7Congruent triangles..................................................................................................8Lecture 4–Wednesday, September 11, 2019................................................................8Subdividing polygons..................................................................................................8Regular polygons.........................................................................................................8Tiling floor with regular polygons.............................................................................9Lecture 5–Friday, September 13, 2019.......................................................................10Parallels.....................................................................................................................10Euclid’s five axioms...................................................................................................10Lecture 6–Monday, September 16, 2019....................................................................11The most famous theorem: Pythagorean theorem....................................................12Lecture 7–Wednesday, September 18, 2019..............................................................13Lecture 8–Friday, September 20, 2019.......................................................................14Centers of a triangle..................................................................................................15Circumcenter.........................................................................................................15Lecture 9–Monday, September 23, 2019....................................................................16Incenter..................................................................................................................16Centroid.................................................................................................................16Lecture 10–Wednesday, September 25, 2019............................................................17Quiz 1 review (Part 1)–Monday, September 30, 2019.................................................17Homework 3, #13......................................................................................................17About the quiz...........................................................................................................18
2©2019 An Li. All rights reserved.How to draw a bisector with a compass?.................................................................18Quiz 1 review (Part 2)–Wednesday, October 2, 2019..................................................18Homework 6, #5........................................................................................................19Lecture 11–Monday, October 7, 2019........................................................................19Back to Thales II........................................................................................................19Lecture 12–Wednesday, October 9, 2019...................................................................21Altitude of a triangle...................................................................................................21Lecture 13–Friday, October 11, 2019..........................................................................23Thales (continued).....................................................................................................23Cyclic quadrilaterals...............................................................................................23Back to orthocenter...............................................................................................23Lecture 14–Wednesday, October 16, 2019.................................................................24Lecture 15–Friday, October 18, 2019..........................................................................25Homework 7, Problem 19..........................................................................................25Lecture 16–Monday, October 21, 2019......................................................................26Lecture 17–Wednesday, October 23, 2019.................................................................26Four-dimensional space............................................................................................26Lecture 18–Friday, October 25, 2019..........................................................................27Lecture 19–Monday, October 28, 2019......................................................................28Lecture 20–Wednesday, October 30, 2019.................................................................29Lecture 21–Friday, November 1, 2019........................................................................30Lecture 22–Monday, November 4, 2019.....................................................................31Nine-point circle, revisited.........................................................................................31Lecture 23–Wednesday, November 6, 2019...............................................................32Lecture 24–Friday, November 8, 2019........................................................................33Lecture 25–Wednesday, November 13, 2019.............................................................34Regular polyhedra.....................................................................................................34Lecture 26–Friday, November 15, 2019......................................................................35Recall: The “shadow” of a polyhedron......................................................................35Lecture 27–Monday, November 18, 2019...................................................................37Lecture 28–Wednesday, November 20, 2019.............................................................37
©2019 An Li. All rights reserved.3Lecture 29–Friday, November 22, 2019......................................................................38Ceva’s Theorem........................................................................................................39Lecture 30–Monday, November 25, 2019...................................................................39Back to the nine-point circle......................................................................................39Ceva’s Theorem (revisited)........................................................................................40Menelaus...................................................................................................................40Lecture 31–Wednesday, November 27, 2019.............................................................40Lecture 32–Friday, November 29, 2019......................................................................41Homework 11............................................................................................................41Problem 5...............................................................................................................41Lecture 33–Monday, December 2, 2019.....................................................................42Problem 8...............................................................................................................42Lecture 34–Tuesday, December 3, 2019....................................................................43Recall: Menelaus.......................................................................................................43
4©2019 An Li. All rights reserved.Lecture 1–Wednesday, September 4, 2019HistoryFor 200 years, geometry was done without graphs nor equations.Real mathematicians see a picture when they see an equation.Unfortunately today, it is just the algebra.This course is about thinking math in pictures and not equations.Geometry begins with Greek philosopher Thales (~600 BC).It is possible that not every triangle in the entire world has its angles sum up to 180°.He asked, “How do you know it’s true?”Measuring is not feasible, because firstly, there is an error with measuring instrumentsand we cannot simply measure every single triangle in the world.
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