Discussion of HW13
a) Suppose = 6 points are given, not all collinear.
Must one of the points lie on at least 2 connecting lines? Yes, by the same argument
as given in class for = 5.
Must one of the points lie on at least 3 connecting lines? Yes, again
Discussion of HW2
The dual of the theorem given may be stated as follows: Let U V W be a triangle
inscribed in a circle , and let u, v, w be the tangents to at U , V and W respectively.
Then the three points
U V w, V W u and W U v
are collinear. (The name
Discussion of HW4
Addition and multiplication tables for a field with four elements look like
0
0
0
1
0
0
0
1
1
0
1
0
0
+
0
0
0
1
1
1
Using the fact that 0 + = 1 = and 0 = 0 for all x. The multiplication table completes
uniquely as
0
0
0
1
0
0
0
1
0
1
0
1
Discussion of HW5
We want to determine x, the number of affine planes of order 3 in a standard deck of Set cards.
We may do so indirectly, by counting in two different ways the number of elements in T, the set
of all 4-tuples (A,B,C,P) where A,B,C form a
Discussion of HW6
There are only 49 points (, ) for , so it is possible to check them one at a time to see
which ones satisfy the equation 2 + 2 = 1 for the conic . In each case, we compute the slope
of the tangent line and finally the equation of the tan
Discussion of HW7
Everyone managed to find coordinates for points and lines of the projective plane of order 3.
Here is Sirenas illustration, for example:
We observe that this is a projective plane; but why it works may have been a mystery. In
particular,
Discussion of HW9
Given three distinct collinear points , and X, show how to construct the unique circle which
is orthogonal to every circle through and (including the line , which we think of as a
circle of infinite radius). You may use the fact that if
Discussion of HW10
In one case, the solution is easy: namely, when and are concentric. In this case the rotational
symmetry about the common center shows that it does not matter where the first tangent circle 1
is placed. If we can find an inversion that
Discussion of HW11
As discussed in class, each pair of points P,Q lies in four lines:
This follows directly from the fact that there are 10 points, and any three points determine a
unique circle. In particular the points A,B lie in one extended affine lin
Discussion of HW12
Solution Using Standard Trigonometric Identites
Let x = cos 72 ; then
x = cos 72 = cos 288 = 2 cos2 144 1 = 2(2 cos2 72 1)2 1
= 2(2x2 1)2 1 = 8x4 8x2 + 1
so
0 = 8x4 8x2 x + 1 = (x 1)(2x + 1)(4x2 + 2x 1).
1
The roots ofthis polynomial ar
Discussion of HW1
We use fairly standard notation which denotes by AB the line joining points A and
B ; the line segment joining A and B is denoted AB . In order to dene line segments,
we require a notion of betweenness, which exists in the Euclidean sett