Geometric Constructions

If $a$ and $b$ are constructible numbers and $a \geq b > 0$, give a geometric proof that $a + b$ and $a - b$ are constructible.

If $a$ and $b$ are constructible, give a geometric proof that $ab$ is constructible. (Hint: Consider the following figure. Two segments of lengths $1$ and $b$ are marked off along a common base line from a common left endpoint, with $b > 1$; from that same endpoint a second ray is drawn at an angle to the base line, and a segment of length $a$ is marked off along it. The segment from the tip of the length-$1$ mark on the base line to the tip of the length-$a$ segment on the second ray is drawn, and then the line through the tip of the length-$b$ mark parallel to that segment is drawn; it meets the second ray (extended) at the point that is distance $ab$ from the common endpoint. Notice that all segments in the figure can be made with an unmarked straightedge and a compass.)

Prove that if $c$ is a constructible number, then so is $\sqrt{|c|}$. (Hint: Consider the following semicircle with diameter $1 + |c|$: mark off a segment of length $1$ and, continuing along the same diameter, a segment of length $|c|$; erect a perpendicular segment of length $d$ from the point dividing them up to the semicircle, and join that point on the semicircle to both ends of the diameter. This creates two right triangles sharing the segment of length $d$ as a leg: one with legs $1$ and $d$, the other with legs $|c|$ and $d$. The angle at the left end of the diameter (in the triangle with leg $1$) and the angle at the apex on the semicircle between the segment of length $d$ and the hypotenuse to the right end of the diameter (in the triangle with leg $|c|$) are both equal to $\alpha$.) (This exercise is referred to in Chapter 31.)

If $a$ and $b$ ($b \neq 0$) are constructible numbers, give a geometric proof that $a/b$ is constructible. (Hint: Consider the following figure. From a common point labeled $0$, the base line carries a mark at distance $a$; a second ray through $0$, at an angle to the base line, carries a mark at distance $1$ and, continuing along that same ray, a mark at distance $b$. The segment from the length-$b$ mark on the second ray to the length-$a$ mark on the base line is drawn, and then the line through the length-$1$ mark parallel to that segment is drawn; it meets the base line at the point that is distance $a/b$ from $0$.)

Prove that $\sin \theta$ is constructible if and only if $\cos \theta$ is constructible.

Prove that an angle $\theta$ measured in degrees is constructible if and only if $\sin \theta$ is constructible.

Prove that $\cos 2\theta$ is constructible if and only if $\cos \theta$ is constructible.

Prove that $30^{\circ }$ is a constructible angle.

Prove that a $45^{\circ }$ angle can be trisected with an unmarked straightedge and a compass.

Prove that a $40^{\circ }$ angle is not constructible.

Show that the point of intersection of two lines in the plane of a field $F$ lies in the plane of $F$.

Show that the points of intersection of a circle in the plane of a field $F$ and a line in the plane of $F$ are points in the plane of $F$ or in the plane of $F(\sqrt{\alpha })$, where $\alpha \in F$ and $\alpha$ is positive. Give an example of a circle and a line in the plane of $\mathbb {Q}$ whose points of intersection are not in the plane of $\mathbb {Q}$.

Prove that $8x^3 - 6x - 1$ is irreducible over $\mathbb {Q}$.

Use the fact that $8\cos^3(2\pi /7) + 4\cos^2(2\pi /7) - 4\cos (2\pi /7) - 1 = 0$ to prove that a regular seven-sided polygon is not constructible with an unmarked straightedge and a compass.

Show that a regular 9-gon cannot be constructed with an unmarked straightedge and a compass.

Show that if a regular $n$-gon is constructible, then so is a regular $2n$-gon.

(Squaring the Circle) Show that it is impossible to construct, with an unmarked straightedge and a compass, a square whose area equals that of a circle of radius $1$. You may use the fact that $\pi$ is transcendental over $\mathbb {Q}$.

Use the fact that $4\cos^2(2\pi /5) + 2\cos (2\pi /5) - 1 = 0$ to prove that a regular pentagon is constructible.

Can the cube be ``tripled''?

Can the cube be ``quadrupled''?

Can the circle be ``cubed''?

If $a$, $b$, and $c$ are constructible, show that the real roots of $ax^2 + bx + c$ are constructible.