Math 5-335 Section 2
Fall 2005
Hints and
supplementary exercises for the October 20 homework
Hint for §4.11, problem 35 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 14 of Chapter 2 [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, there is yet one more issue.
Namely, we have to apply Proposition 13 of Chapter 2 [about addition
of angular measure] in order to give a full explanation of equation (4.1)
in the proof of Theorem 2. In the application of that Proposition 13,
one needs to check that A is in
the interior of angle
Supplementary problem #1
Given triangle
Supplementary problem #2 Consider the triangle shown
in the following figure:
Comments and questions top;
roberts@math.umn.edu
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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.
ABC,
let P be the midpoint of
, let
Q be the midpoint of
, and let
R be the midpoint of
.
is parallel to
.
Suggestion: Using appropriate coordinate vectors for
P and Q, find a direction
indicator for
and compare it with one of the usual direction indicators for
.
is parallel to and that
is parallel to
.
| =
1/2 ||
| =
1/2 ||
| =
1/2 ||
ABC is equal to the
orthocenter of triangle
PQR.
Please note: If we haven't covered both of the
pertinent sections, then you'll have to look up the definition of
orthocenter in §4.4 and the definition of
circumcenter in §4.6, but you
should not need very much else from those sections.
In particular, this problem is designed to be workable without
use of the formulas from those sections.
Further suggestion: Suppose that we draw the perpendicular
bisector of a side of triangle
ABC. In what role
does the line that we drew function relative to triangle
PQR?
( This is intended to be a geometry
problem. In particular, it is recommended not to use
the formulas given in Theorem 8, Corollary 11, and Theorem 14 of
Chapter 4.)
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.