MATH 427, Fall 2026
FOUNDATIONS OF GEOMETRY
(Euclidean and various non-Euclidean geometries
and their development from postulate systems.)

Syllabus - Extra credit problems - Tickets

• HWA-11, due 11:15, Mon, Nov 9.
  • Read and understand: Chapter 11.
  • Read and think: Chapter 12.
  • Exercises: 11.2, 11.4, 11.7, 11.11, 11.14.
• HWA-10, due 11:15, Mon, Nov 2.
  • Read and understand: Chapters 10.
  • Read and think: Chapter 11.
  • Exercises: 10.3, 10.8, 10.13.
  • Constructions:
    • Angle-median-side,
    • Inversions,
    • Perpedicular circle 1,
    • Perpedicular circle 2.
• HWA-09, due 11:15, Mon, Oct 26.
  • Read and understand: Chapter 9.
  • Read and think: Chapter 10.
  • Exercises: 9.6, 9.7, 9.25.
  • Constructions:
    • Equilateral triangle,
    • Perpendicular with ruler,
    • Circumradius-inradius-side,
    • Inversion.
• HWA-08, due 11:15, Mon, Oct 19.
  • For full credit, do 10 of the following 12 constructions:
    • ⅜π,
    • 2:3,
    • Parallel line,
    • Bisection 3,
    • Reflection 2,
    • Tangent circle 1,
    • Equilateral triangles,
    • Concentric circles 1,
    • Concentric circles 2,
    • Minimal perimeter 1,
    • Inscribed eqilateral triangle 1,
    • Vertex of bisected angle.

• Midterm, Fri Oct 16, in class, 5 problems:
  • 2 theorems from the list (possibly with related definitions):
    2.4, 2.7, 2.12, 2.14,
    3.6, 3.9, 3.10, 3.13,
    4.1, 4.2, 4.4,
    5.2, 5.6, 5.8, 5.13, 5.20, 5.22,
    6.6,
    7.1, 7.2, 7.8, 7.9, 7.13, 7.18,
    8.1, 8.3, 8.6, 8.11, 8.15, 8.16;
  • 2 problems from HWA 01--07 (possibly with related definitions);
  • one new problem.
• You may prepare and use a cheat sheet --- one sided standard paper size, include diagrams and formulas, but no words. Do not make it cryptic.
• I will print for you the list of axioms, plus you may ask for something else (in advance).

• HWA-07, due 11:15, Mon, Oct 12.
  • Read and understand: Chapter 8.
  • Exercises: 7.23, 8.5, 8.7, 8.14, 8.17.
  • Start to think about midterm exam --- skim thru the covered material, prepare at least one question.
• HWA-06, due 11:15, Mon, Oct 5.
  • Read and understand: Chapter 7.
  • Read and think: Chapter 8.
  • Exercises: 6.10, 7.10, 7.21.
  • Constructions:
    • Inscribed square 1,
    • Segment trisection,
    • Tangent circle 3,
    • Inscribed triangle.
• HWA-05, due 11:15, Mon, Sep 28.
  • Read and understand: Chapter 6.
  • Read and think: Chapter 7.
  • Exercises: 5.23b, 6.3, 6.11.
  • Constructions:
    • Tangent circle 2,
    • Bisection 2,
    • Circle thru point tangent to lines 2,
    • OX·OZ = OY².
• HWA-04, due 11:15, Mon, Sep 21.
  • Read and understand: Chapter 5.
  • Read and think: Chapter 6.
  • Exercises: 5.3, 5.4, 5.21.
  • Constructions:
    • Center of circle,
    • Tangent circle 3,
    • Minimal sum 1,
    • Tangent line 2.
• HWA-03, due 11:15, Mon, Sep 14.
  • Read and understand: Chapters 3 and 4.
  • Read and think: Chapter 5.
  • Exercises: 3.14, 3.15, 4.3.
  • Constructions:
    • Copy of angle,
    • Side-median-side,
    • Angle-side-side,
    • Bisected line.
• HWA-02, due 11:15, Fri, Sep 4.
  • Read and understand: Chapter 2.
  • Read and think: Chapter 3.
  • Exercises: 1.15(b), 1.18, 2.5, 2.10.
  • Register at geogebra.org and do the following constructions.
    • Side-angle-side,
    • Motion.
    (Once you see "Well done", make a screenshot and upload it to CANVAS.)
• HWA-01, due 11:15, Fri, Aug 28.
  • Read and understand: Chapter 1.
  • Read and think: Chapter 2.
  • Exercises: 1.2, 1.4, 1.11, 1.13, 1.15(a).
    (Scan to pdf and upload to CANVAS.)

Extra credit:

Exercises: 1.8, 3.17, TBC.

Do the construction and prove that it works:

• Minimal perimeter 2,
• Vertices on lines.
• Altitude-bisector-median.
• Perpendicular circle 3.

0. Finding a mistake or misprint in the book earns a tiny bit of extra credit, depending on the nature of the mistake.

1. Show that every 4-point metric space is isometric to a subset of the Manhattan plane \((\mathbb{R}^2,d_1)\).

2. Let \(f\colon S\to\mathbb R\) be a distance-nonexpanding function defined on a subset of a metric space \(\mathcal X\); that is, suppose that \(|f(A)-f(B)|\leqslant AB\) for all \(A,B\in S\). Show that \(f\) can be extended to a distance-nonexpanding function defined on the whole \(\mathcal X\).

3. Describe all lines in the Manhattan plane \((\mathbb{R}^2,d_1)\). Namely, show that a subset \(\ell\) of the Manhattan plane is a line if and only if it has a unique point of intersection with every graph \(y=m\cdot x+b\) with \(m>0\), or with every such graph with \(m<0\).

4. Let \(H_4\) be the 4-dimensional Hamming cube; that is, the set of all sequences of four zeros and ones, with the distance defined as the number of positions in which two sequences differ. (For example, \(\mathrm{dist}(0101,0011)=2\) since the sequences differ in exactly the second and third positions.) Show that any two isometric 3-point subsets \(S\) and \(S'\) of \(H_4\) are congruent; that is, there is a motion of \(H_4\) that maps \(S\) to \(S'\). Show that the analogous statement fails for 4-point subsets.

5. On several occasions, hordes of zombies entered a house. Each time, the zombies scattered thruout the house and went dormant. The next morning, a student chooses a place in the house and sits still; all the zombies start moving toward him to eat his brain. Prove that on each occasion the student could choose his place so that the average distance the zombies would have to travel was the same on every occasion, regardless of how many zombies there were or where they were. Assume that the house did not change. Distance is measured by the length of a shortest path, and such a path exists between any two points of the house.

6. Show that a composition of three reflections across the sides of a nondegenerate triangle does not have a fixed point.