Math 5-335 Fall 2004
Hints and
supplementary exercises for the October 21 homework
Supplementary problem #1 Consider the triangle shown in the following figure:
that is closest to the incenter, and find the inradius
(radius of the inscribed circle).
and 
that are closest to the incenter. Label all 3 of these points of tangency
on a copy of the diagram.Hint for §3.12, problem 64 Here's a diagram of the setup:
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.
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 A is in
the interior of angle
BCE. Showing
that A and B are on the same side of
k is part of this,
but not the whole story. Indeed, it also involves
showing that A and E 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.
Comments and questions top;
roberts@math.umn.edu
Back to to the homework list.
Back to the class homepage.