MTHT 411 Advanced Euclidean Geometry

Spring Semester, 2009

Instructor: Anatoly Libgober
Office: SEO 413
Phone: 312-413-2138
Office Hours: MW 10 a.m. or by appointment.
Text:  William Barker and Roget Howe, Continuous Symmetry: from Euclid to Klein, AMS 2007.
Hours: The course meets Thursdays, 5-8 p.m. at 207 Taft Hall

Topics

This course gives an introduction to modern Euclidian Geometry. We shall discuss axiomatic development of Eucledian Geometry, the role of transformations in Euclidian Geometry (in particularly isometries and similarities). It includes extensive development of problem solving skills, applications to physics and comparisons between Euclidian and non-Euclidian Geometries. We shall try to cover most of the material in the text.  

Grading and Homework

Grades will be based on midterms and homework problems assigned regularly. Problems will be assigned during classes or posted on this webpage.

Other Sources

E.E.Moise, Elementary Geometry from Advanced View Point. Addison-Wesley, Reading, MA, 1990.
H.S.M. Coxeter and S.L.Gretzer, Geometry Revisited, The mathematical associatioin of America, Wahington, D.C, 1967.
F.Klein, Elementary Mathematics from Advanced Standpoint, Dover Publications, 1945.

Homework 1: Read Section I.1. Exercises 1.1(b),1.2 (a),(b),(c), 1.3,1.4,1.6.
Read section I.2 except examples 2.4 and 2.9. Exercises 2.1, 2.2,2.3,2.5,2.6. Solutions to problems from both sections are due Jan 22.
Homework 2:Read examples 2.4 and 2.9. Exercises 2.4 (b)-(f), 2.7. Problem: Give an example of two lines L(a,b,c) and L(a',b',c') on projective plane which coincide but have different (a,b,c) and (a',b',c'). Find a relation between a,b,c,a',b',c' which if satisfied implies that two lines with different (a,b,c) and (a',b',c') coincide. Exercise VI.2.1. Problem: Describe a construction of circumcenter using ruler and compass.
Homework 3:Chapter I: Exercises 3.1,3.2,3.3,3.4. Chapter VI: Exercise 2.2(a). Problem: Suppose that the area of a triangle is 1. What is the area of its medial triangle?
Homework 4:Chapter I: Exercises 4.1,4.2,4.4 Problem 1: For the dilation by (-2) with the center at centroid of a triagle, describe transforms of the vertices of medial triangle. Problem 2: Describe composition of two reflections in perpendicular axes.
Homework 5: Exercsies 5.1,5.6,5.9. (due. Feb 26). Topics for the first midterm: Chapter I, sections 1 through 5. Chapter VI, sections 1,2 and 3.1.
Midterm 1 (.pdf)
Homework 6: (due March 5) Exercese 6.2 p.43. Problem 1. Show that signed uniform dilation transforms lines into line. Problem 2. Show that signed uniform dilations preserve angles. Problem 3. Find a relation between the length of a segment and its dilation. Find the relation between the area of a triangle and its dilation. Do Exercise 3.2 p. 306. Study sections 3.8-3.12.
Homework 7: (due March 12) Write down a proof of existence of nine point circle. Exercise 3.3. Prove (a) of proposition 3.18.
Homework 8: (due March 19) Exercises: Chapter I, 7.2, 7.3(b). Chapter VI: Exercise 3.5.
Homework 9: (due April 2) Exercises: Chapter I, 8.2, 8.3, 8.4,8.5. Write down a proof showing that triangle inequality does not follow from the axioms of sections I.1-I.6 (i.e. cannot be proven without SAS axiom).
Homework 10: (due April 9) Exercises: Chapter I, 9.1, 9.4. Write down a proof showing that composition of two reflections is either translation or rotation. Exercises 2.1, 2.2 and 2.7 on p. 186. Midterm 2 will take palce on April 16.
Midterm 2 (.pdf)
Homework 11: (due April 30) Exercises: Chapter III Exercises 2.3, 2.4, 2.6, 2.8, 2.9 on p. 186. In solution to Fagnano problem the perimeter of inscribed triangle is identified with the length of a broken line consisting of three segements. For which inscribed triangles this broken line a) consists of two segements b) consists of one segement.
Final Exam: May 7, 5-7 p.m.