Mathematics 138
Problem Set 2 due Wednesday, February 23, 2005 
Reading: attached notes on three ways to measure angles and lengths.
Yaglom, pages 214-255 .  You'll have to read this several more times before it makes sense.
1. Suppose that C and E are members of a committee of an affine faculty senate whose additive identity is O and whose multiplicative identity is I.  Describe a method for constructing the instructor A such that AC = E using auxiliary member B.  Prove directly from the axioms that using a different member B'  on the same committee as I and B leads to the same A. (Of course, this is true for arbitrary B', but the proof becomes tedious).Include a diagram to illustrate how your construction would look in the Euclidean plane, for which your proof is also valid.
2. Consider the large affine faculty senate. Choose the vacation committee as the x axis and the gifts committee as the y axis.  Betty, who serves on both these committees, has coordinates (0,0).  Let Maria be the multiplicative identity on the vacation committee, and let Quint be the multiplicative identity on the gifts committee. 
All coordinates are in the field Z5.
a. Determine what committee satisfies x = y, and assign coordinates to all its instructors.
b. What instructor has coordinates (1,2)?
c. What are the coordinates of Oscar in this coordinate system?
d. What committee satisfies the equation y = 3x + 2?
e. What equation is satisfied by the diversity committee?
3. Consider the medium affine faculty senate.  Choose the housing committee as the x-axis, with Lynn as multiplicative identity.  Choose the medical committee as y-axis, with Adam as multiplicative identity.
The finite field F4 has elements 0, 1, u, u+1, where u(u+1)=1, and all arithmetic is carried out modulo 2; i.e. 1+1 = 0.
a. What instructor has coordinates (0,0)? What instructor has coordinates (1,1)?  What committee satisfies equation y = x? Assign coordinates (u,u) to whichever remaining instructor on this committee is first alphabetically.
b. Determine the coordinates of all the instructors. Show your answer on a diagram, with the
x axis horizontal and the y axis vertical.
c. Which committee satisfies y = ux, and which satisfies y=(u+1)x?
d. What equation is satisfied by the tenure committee?
e. What committee satisfies the equation x + (u+1)y = u?
4. Consult Appendix F of Ryan to review (or learn) hyperbolic functions.  Now define 
tanh u = sinh u/cosh u.
a. Find a formula for tanh (u + v) in terms of tanh u and tanh v.
b. Sketch a graph of tanh u. 
c. If x = tanh u, then u = arctanh x.  Sketch a graph of arctanh x and find a formula for its derivative.
d. What is the relationship between the function arctanh m and the function
that arose in conjunction with hyperbolic angles? (Hint, differentiate both and compare the values for m = 0.)

5. Consider two lines whose slopes v and w are both less than 1 in magnitude.  Think of these as graphs of the motion of two relativistic particles whose velocities are v and w respectively in a system of units where the speed of light c = 1.
a. What hyperbolic angle does each line make with the x axis?  What, therefore, is the hyperbolic angle between the lines?
Let u denote the velocity of the second particle in a frame of reference where the first is at rest.  According to Einstein,
. Show that the hyperbolic angle between a line of slope u and the x axis is equal to the hyperbolic angle that you calculated in part a.

6. Consider two lines whose slopes are v and w.  If you perform a Euclidean rotation that carries the first line into the x axis, what is the new slope u of the second line as a function of v and w? This formula is the analogue of Einstein's velocity transformation formula in the case where elliptic measure of angles is used. 
7. The Law of Sines sometimes provides an easy way to prove theorems without constructing any extra lines.  In the cross-ratio proof done in lecture the trick was that sines of supplementary angles are equal and cancel.  A similar trick is that vertical angles are equal and their sines cancel. Use it to prove the famous Theorem of Menelaus:

Hint: Apply the Law of Sines to CFE, BDE, and DFA, using it to relate angles at D, E, and F to the opposite sides.  Then combine the resulting equations so that all the sines cancel out. Note that sin CFD = sin AFD

8. Prove the theorem of Ceva:
               Hint: apply the Law of Sines to the six triangles with a vertex at O, and arrange for all the sines and the lengths OA, OB, OC to cancel out.

9. One of the most remarkable theorems of real affine geometry is illustrated by this diagram:

Points B, J, and F trisect the sides of the triangle. The theorem states that the area of EHG is 1/7
of the area of ABC. I learned this theorem when taking Physics 232 in 1962, but I did not know how to prove it until I mentioned it in Math 191 and a student (Andrew Chi, I think) pointed out that it can be proved using the theorem of Menelaus.
This is a genuine affine theorem, since any affine transformation of the plane (linear transformation
followed by translation) preserves ratios of areas and ratios of lengths of segments on the same line.
Let Z denote the area of triangle ADC.
a. Prove that the area of CBD (and of two other triangles) is Z/3. Use the theorem of Euclid about areas of triangles with the same altitude and different bases on the same line (Book VI, Proposition 1).
b. Use Menelaus to prove that CE = 6 EB.  In the diagram, BC appears nearly perpendicular to AD, but that is an accident of the diagram.
c. Prove that the area of EBD (and of two other triangles) is Z/21.
d. Prove that the area of EHG is Z/7.  There are lots of equivalent ways to do this.