Math 403: Euclidean Geometry
Midterm 2 - Review

Time & Location: Thursday, Mar. 17, 10:00-10:50, 441 Altgeld.
Material: Sections 2.1-3.3 of the textbook.

You should be prepared to state any of the following

Axioms:

• Axioms (SP1)-(SP4) for the dot product.

Definitions:

• Translation τ_A
• Central dilations δ_r and δ_C,r
• Central reflection σ_C
• Group, subgroup
• Homomorphism of groups
• Isomorphism of groups
• Abelian group
• Cyclic group
• Generator of a (cyclic) group
• Order of an element in a group
• Order of a group
• Dot product (or inner product, scalar product)
• Length of a vector and distance between points <
• Orthogonal vectors
• Rhombus, rectangle
• Perpendicular bisector, altitude, foot of an altitude
• Circumcenter, Orthocenter
• Euler line, Euler circle

Results:

• The conjugation formula for dilations (equation 14)
• Snapper's Theorem
• Proposition 2.21
• Thales' Theorem
• Nine-point Circle Theorem

In addition, you should know what you get by composing dilations. For instance, a composition of two reflections is a translation (Theorem 2.15), and τ_Aδ_C,r is again a central dilation if r is not 1 (formula 30). You do not need to memorize things like formula 30, but you should know that the answer is some central dilation.
You should also be familiar with all of the examples of groups we studied, like the cyclic groups Z/nZ, the Klein 4-group K, and the dihedral groups D_n. This includes the representation of elements of these groups as permutations of the vertices.

Suggested Exercises: Try the odd-numbered exercises in Chapter 2 and 3, especially problems 2.3, 2.5, 2.9, 2.21, 2.25, 3.1, 3.5, 3.9 Some of these were done in class, and they all have hints in the back of the book, but try to do them without consulting your notes or the back of the book. Another good thing to try is to look at some of the results in the text (like Proposition 2.21 or 3.1, for example) and try to prove them without looking at the proof in the book.