Download HW6 - Harvard Math Department

Mathematics 138
Problem set 6 due Monday, March 21, 2005
The midterm exam on March 23 will cover material up through this problem set. There will be
more Galilean geometry on Monday the 23rd, but it won't be on the exam.
1. Yaglom, problem 9 on pp. 65- 66. You are to construct a proof of the dual theorem, not just to
invoke the principle of duality. The dual of a dilatation is a compression, as was mentioned briefly in
lecture in conjunction with SSS congruence. A compression is a change of scale along the vertical
direction only.
2. Yaglom, problem 11 on page 66. You are not required to prove the theorems. Just state them
and draw accurate diagrams to illustrate them.
3. Yaglom, problem 12 on page 66. Yaglom's hint is one way to construct a proof. Another is to
view this as a special case of Ceva's theorem (see problem 2). The diagonals of the trapezoid
and the median of the triangle are the three concurrent lines. In Yaglom's diagram, QD/AD =
QC/AC, and Ceva's theorem says that AM = BM. The standard proof of Ceva's theorem uses the
Euclidean law of sines, but it works also with the Galilean law of sines a/A = b/B = c/C. Just
dualize it for this special case.
4. A well-known theorem of Euclidean geometry is that the diagonals of a rhombus are
perpendicular. (A rhombus is defined as a parallelogram all of whose sides have equal length.)
a. Formulate and prove a Galilean version of this theorem, and draw a diagram to illustrate it.
b. Formulate and prove the Galilean dual of the theorem from part a, and draw a diagram to
illustrate it. You will have to define a co-rhombus as a special sort of co-parallelogram.
5a. In Euclidean geometry, each diagonal of a rhombus bisects the angles at the two vertices that it
joins. Formulate and prove a Galilean version of this theorem, and draw a diagram to illustrate it.
b. Formulate and prove the Galilean dual of the theorem from part a, and draw a diagram to
illustrate it.
6. Here is a physics problem.
"At time t = 0, Bill boards a train in Union Station and travels north at velocity v 1 for time t1. He then
catches a southbound train and travels at velocity v 2 (a negative number) for time t2 - t1, returning
to Union Station at time t2.
So v1t1 + v2 (t2 - t1) = 0.
Paul is riding a slow train traveling south with velocity V (a negative number). He passes Bill at time
T1 as Bill is heading north, then again at time T 2. as Bill is heading south. He finally reaches Union
Station at time T.
Prove that T(t2-T2)(t1-T1) = T1 (T -t2 ) (T2 - t1)."
a. Draw a diagram to illustrate this. A good set of values is T 1 = 2, t1=3, T2=6, t2=8, T = 12.
b. Do the proof by identifying this problem as the Galilean version of a famous theorem that you
proved earlier in the course and making the obvious change to the Euclidean proof, which uses the
Law of Sines.
c. Draw a diagram to illustrate the Galilean dual of this theorem. You do not need to prove it. Just
for fun, you could state it as a physics problem, but this is not required.
7a. A Galilean isogonal (and hence also isosceles) triangle abe is converted into a co-trapezoid by
constructing lines c and d that join vertices P and R of the original triangle to the midpoints M and N
of the opposite sides. These lines meet at S, which becomes the fourth vertex of the cotrapezoid.
Theorem T states that the angle ab at Q is three times as large as the angle dc at S.
State the Galilean dual of theorem T, and draw a labeled diagram to illustrate it. To avoid giving
misleading Euclidean impressions, include no horizontal lines in this diagram.
b. Prove both theorem T and its Galilean dual. Start with whichever is easier. You may use results that
were proved in class, including the Galilean triangle equality and its dual and the Galilean Law of Sines.
Q
b
a
N
M
S
c
d
P
e
R
8a. Convert theorem T from the preceding problem into a "physics problem." It might start like this:
"Huey, Louis, and Dewey are all traveling from event P (time 0) to event R (time 2T). Huey rides
train e south to make the journey. Louie rides train c north until time T, then hops onto train d to
finish the journey. Dewey rides train b north..." You will have to work the midpoints into the story
line and state the conclusion of the theorem (to be proved) in physical terms.
b. Convert the dual of theorem T into a physics problem. Here is a possible start:
"A family of bird fanciers is traveling north in two trucks q and s, which move at the same velocity,
with s north of q. At time 0, the riders in q dispatch a fast pigeon p north to send a message to s,
and they also send a slow crow m whose velocity is the average..." You have to work in the angle
bisectors and the fact that the trapezoid is isosceles, and state the conclusion of the theorem in
terms of when the fast pigeon reaches truck s and when pigeon q makes the return trip.
Once you have disguised this pair of theorems as physics problems, their duality will not be
apparent.