Math 403: Euclidean Geometry
Midterm 3 - Review

Time & Location: Tuesday, Apr. 19, 10:00-10:50, 441 Altgeld.
Material: Sections 3.4-4.10 of the textbook (from section 4.10, you only need to know the main theorem, Theorem 4.36).

You should be prepared to state any of the following

Axioms:

• Properties (D1)-(D3) of a distance function

Definitions:

• Distance between X and Y
• Unit vector
• Orthogonal projection of Y onto the vector X
• Orthogonal projection of Y onto an arbitrary line l
• Cosine formula for angles (formula (23))
• Isometry, linear isometry
• Reflection through a line l_A,B
• Rotation
• Glide reflection

Results:

• Cauchy-Schwarz inequality
• Law of cosines (formula (24))
• Isometries preserve angles and preserve lines
• Isometries form a group
• Linear isometries preserve dot products
• Conjugation law for reflections (4.13) and for central reflections (4.17)
• Every isometry can be written as a product of up to three reflections
• A composition of two rotations is either a rotation or a translation.

In addition, you should know how isometries are determined by their fixed points:

• If an isometry fixes X and Y, then it fixes the entire line l_X,Y. If this isometry is not the identity, it must be the reflection σ_lX.Y.
• If an isometry fixes three noncollinear points X, Y, and Z, then it must be the identity.
• If an isometry fixes only the point X, it must be a rotation ρ_X,θ
• If an isometry has no fixed points, it must be a translation or a glide reflection.

and how many reflections are needed to express a type of isometry:

• Reflections already are expressed as a single reflection
• Translations, rotations, and central reflections can be expressed as a composition of reflections through two lines l and m. For translation, the lines are parallel. For rotations, they intersect at the rotation point C, and the angle between the lines l and m is θ/2. For central reflections, they are orthogonal and intersect at the reflection point C.
• Glide reflections can be expressed as a composition of three reflections.

Suggested Exercises: Try the odd-numbered exercises in Chapter 3 and 4, especially problems 3.11, 3.13, 4.1, 4.5, 4.13, 4.17, 4.21. 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 4.2, 4.7, or 4.14, for example) and try to prove them without looking at the proof in the book.