Math 5-335 Fall 2003
Hints and
supplementary exercises for the October 16 homework
Supplementary exercise #1 Consider the same triangle ABC as in problem 30 of §2.5.
= -A +2B =
B - (A - B)
= -A +2C =
C - (A - C).
(By Theorem 16 in Chapter 2, these points should be on the lines
and
respectively.
is on this line if and only if r = -1.
and (a,0,-a+1).
Supplementary exercise #2 Consider the same triangle ABC as in problem 30 of §2.5.
(which lies on the line
).
of this line,
the equality of ratios
s/t =
1/2 holds.
is a point of the line
, then for every point
(r,s,t) of the
line joining A and P, the equality
of ratios
s/t =
b/c holds.
Hint for §2.5, problem 30 In the first part of the
problem, we're supposed to plot the line consisting of points whose points
have barycentric coordinates (r,s,t) where
r has a constant value. We can draw such a line
by finding two points on it. The two easiest such points to find
are the ones where the line r = (constant)
intersects the lines
Hint for §3.12, problem 64
(Re-written retroactively)
Here's a diagram of the setup:
Now, the text says that the only other loose end is about showing that
A and B are on the same side of
k. Actually, in the application of
Proposition 33 of Chapter 1 [about addition of angular measure],
one needs to check that B is in
the interior of angle
Comments and questions top;
roberts@math.umn.edu
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and
. This was done in
Supplementary exercise #1. You don't need to use the formula here; it's
good enough just to solve on a case-by-case basis, noting that the lines
and
are characterized
by the equations
t = 0 and s = 0
respectively. Lines of the form s = (constant)
and t = (constant) are handled similarly.
and further
that A and D are on opposite sides of
while
B and E are on opposite sides of
. Under this setup, we need to
prove that 
and
are opposite rays, so that
we can apply Proposition 34 of Chapter 1 [about complementary angles].
We can reformulate the question about opposite rays as being about
whether D - C and E - C are
negative scalar multiples of each other. Since we're given that
k is parallel to
, it is possible to
compare either D - C or E - C
with B - A, determining
in each case whether the vector in question is a scalar multiple of
B - A, with a positive coefficient or with a
negative coefficient.
ACD. Showing
that A and B are on the same side of
k is part of this. But it also involves
showing that B and D are on the
same side of .
What was proved about oppositeness of rays in the other part of the
problem can be very helpful here.