Section 2: 2.8
Section 6: 6.1, 6.3(c)

A sample solution.

Section 7: 7.1(a), 7.2, 7.4.
(Read and think about Exercise 7.9. You do not have to turn it in.)

Hint for 7.1(a): To show that $(A*B*C)\wedge (B*C*D)$ implies $A*B*D$, consider the following steps. Assuming $(A*B*C)\wedge (B*C*D)$, explain why

Section 7: 7.1(a), 7.2, 7.4.
(Read and think about Exercise 7.9. You do not have to turn it in.)

Hint for 7.1(a): To show that $(A*B*C)\wedge (B*C*D)$ implies $A*B*D$, consider the following steps. Assuming $(A*B*C)\wedge (B*C*D)$, explain why

The main purpose of this assignment is to show that, in any plane satisfying the incidence and betweenness axioms, if you know the betweenness relation on one line, then you can figure out the betweenness relation throughout the whole plane.

Exercise 1.
If $\angle BAC$ is an angle, and $D$ is in the interior of $\angle BAC$, then all points of the ray $\overrightarrow{AD}$, except $A$, are in the interior of $\angle BAC$.

Exercise 2.
(a) Assume that $A, B, C$ are distinct and collinear, and all three lie on the same side of line $\ell$. Explain why there is a point $D$ on the opposite side of $\ell$. Explain why there are distinct points $A', B', C'$ on $\ell$ such that $A*A'*D, B*B'*D, C*C'*D$ hold. Show that $A*B*C$ holds if and only if $A'*B'*C'$ holds.
(b) Assume that $A, B, C$ are distinct and collinear, and that $A$ and $B$ are on the same side of line $\ell$, but $C$ is on the opposite side. Show that there is a point $D$ on the same side of $\ell$ as $C$ such that $DA, DB$ and $DC$ all intersect $\ell$, say at $A', B', C'$. Show that $A*B*C$ holds if and only if $A'*B'*C'$ holds.

Exercise 3.
Show that if we know the betweenness relation for all triples $A', B', C'$ of distinct points of some fixed line $\ell$, then we can determine the betweenness relation throughout the whole plane.

This assignment is about the construction of weird models.

Section 8: 8.7, 8.10.
Section 9: 9.3.

This is a writing assignment. You worked on these problems in class, so here you are just writing down the answers.

Exercise 1.
Explain why if $\ell$ is a line that meets distinct lines $m$ and $n$ in right angles, then $m$ and $n$ are parallel.

Exercise 2.
Explain why if $\ell$ is a line and $A$ is any point (possibly on $\ell$, possibly not), then there is a line $m$ incident to $A$ that meets $\ell$ in a right angle.

Exercise 3.
Suppose that $A$ is not incident to $\ell$. Explain how to construct a line parallel to $\ell$ through $A$.

Let $\Gamma$ be a circle centered at $O$ with radius $\overline{OA}$. Show that a line perpendicular to $OA$ at $A$ is tangent to $\Gamma$.

This assignment is about ordered fields.

Section 13: 13.7 (only answer the first two questions, which are the ones starting ``Find which …'' and ``Can you …'').

Section 15: 15.2.

Exercise 3. Show that $\sqrt{1+\sqrt{2}}$ satisfies a nonzero integer polynomial of degree four, but does not satisfy a nonzero integer polynomial of degree less than four. (Comments: (i) ``Integer polynomial'' means ``polynomial with integer coefficients'', (ii) This problem is related to Exercise 16.10(d), but you do not have to work out that exercise.) Sample solution.

Hint for 16.11: Start with a failure of the circle-circle intersection property, then think about the triangle whose vertices are the two circle centers and the expected point of intersection.

Last Assignment!

We showed that problem (i) is insolvable in general by producing an example of a circle with constructible radius of length $r$ that has the same area as a square with nonconstructible side length $s$. We showed that problem (ii) is insolvable in general by producing a cube with constructible side length $s$ such that the cube with twice the volume has nonconstructible side length $t$. We showed that problem (iii) is insolvable in general by producing a constructible angle $\theta$, such that the angle $\theta/3$ is nonconstructible. (An angle $\theta$ is constructible iff $\cos(\theta)\in K$.)

Exercise 1.
Give an example of a radius of nonconstructible length $r$ such that the circle with radius of length $r$ has the same area as a square with nonconstructible side length $s$, BUT, if you are given $r$, then you can use $r$ to construct $s$.

Exercise 2.
Give an example of a nonconstructible side length $s$ of a cube such that the cube with twice the volume has nonconstructible side length $t$, BUT, if you are given $s$, then you can use $s$ to construct $t$.

Exercise 3.
Give an example of a nonconstructible angle $\theta$, such that the angle $\theta/3$ is also nonconstructible, BUT, if you are given $\theta$ (or its cosine), then you can use $\theta$ to construct $\theta/3$.