Geometry Construction Problems

Question

Recently I've been trying my hand at a few geometrical construction problems using just a straight edge and a compass. So far I have constructed the following:

• an equilateral triangle
• a square
• a regular pentagon
• a circle circumscribed about a triangle
• a circle inscribed in a triangle
• a parallel line through a point
• a perpendicular line through a point
• a bisected angle
• a segment cut into $n$ congruent segments

I can't think of anything else to do other than just regular polygons, and I can't find a good list online. Can anyone think of any other constructions that I could try? I'm new to this, so if you give me something incredibly difficult, I may need a hint.

Thank you!

Answer 1

A good book to consult is "One Hundred Great Problems of Elementary Mathematics: Their History and Solution" (NY: Dover Publications). This is an English-language reprint of a German-language original "Triumph der Mathematik" by Heinrich Dörrie (Leipzig).

1. It has been proven that any construction attainable with unmarked straightedge and compass can be accomplished with compass alone. Try it on some of your already-known constructions. It should be noted that in this mode a straight line is deemed to be known/constructed if two of its points are known/constructed. Section 33 of the source cited above.

2. It has been proven that any construction attainable with unmarked straightedge and compass can be accomplished with straightedge alone, provided that a fixed circle (with center) is present in the vicinity. Try it on some of your already-known constructions. Section 34 of the source cited above.

3. There are constructions which cannot be done with unmarked straightedge and compass that can be done if one is allowed to make two marks on the straightedge (for the purpose of sliding a fixed distance). These are called "neusis" constructions. Try a few of them, e.g., Trisection of an angle; construction of a cube root, construction of angles and regular polygons not constructible by ordinary means, etc.

Answer 2

Try this site, it has 40 different challenges. It might be useful. https://sciencevsmagic.net/geo/

Answer 3

The constructions in your list are the basic (but foundational) ones. Try more complicated variations:

• triangle problems: given 2 angles and 1 side, construct the triangle; given 1 angle and 2 sides; given an appropriate combination of internal lines (altitude, median, angle bisector, ...) and so on (some are more difficult than others);
• "CAD style" constructions: tangents to a given circle from a given point; tangent to two given circles, line tangent to a given circle and perpendicular to a given line, ...
• what arithmetic operations can you construct? (given numbers $a$, $b$, $c$, ... as lengths or segments);
• (points on) curves: parabola, hyperbola, ellipse, catenary, ...
• geometric figures given the area and some other property(ies);
• "minimal" figures: a figure [or one of them] from a certain family that minimises a certain property.

You can also check out all the questions in the "Related" section of this site. :-)

Answer 4

There are interesting straightedge-only constructions as well, particularly in the context of conics.

1. When given four points $P_1,\ldots,P_4$ in general position, these determine three further points: $$\begin{aligned} \{Q_{1234}\} &= \overline{P_1P_2}\cap\overline{P_3P_4} &\{Q_{1324}\} &= \overline{P_1P_3}\cap\overline{P_2P_4} &\{Q_{1423}\} &= \overline{P_1P_4}\cap\overline{P_2P_3} \end{aligned}$$ Now suppose you are given $Q_{1234},Q_{1324},Q_{1423}$ and $P_1$ instead. Find $P_2,\ldots,P_4$. Desargues's theorem helps.
2. Given five points $P_1,\ldots,P_5$ of a conic and an arbitrary point $Q$ in the (projective) plane, it is possible, with the help of Pascal's theorem, to construct the second point of intersection of the conic with the line $\overline{P_1Q}$. Since $Q$ is arbitrary, you can find as many more points on the conic as you like, with only a straightedge needed.

3. Poles and polars with respect to some given conic. For simplicity, begin with a circle for the conic. Given that circle and some point in the (projective) plane (the pole), construct its associated polar, with respect to the circle.

   There are many constructions out there using a compass, but it is possible to use only a straightedge. How? Revisit item (1) in this list. If $P_1,\ldots,P_4$ are on the circle, then $Q_{1234}$ happens to be the pole to the polar $\overline{Q_{1324}Q_{1423}}$. Try to make use of that relation.

One particular benefit of straightedge-only constructions is the following: Since straight lines remain straight under projective transformations, once such a construction works with scenes containing a circle, the construction will work the same way for an arbitrary conic.

Answer 5

Here are a couple more advanced challenges. The construction procedures are not all that complicated but you need a little ingenuity to derive them:

1) Given triangle $ABC $, construct a point $P $ such that $|PA|+|PB|+|PC|$ is minimized. Sometimes this point is just a vertex of the triangle, sometimes it is in the interior. You should be able to identify from the construction which case applies to a given triangle.

2) If you have a sphere available for constructions and a curved "straightedge" matching the curvature of the sphere, construct a regular pentagon of great circular arcs. This involves very different relationships from the planar construction. At least in the way I would do it, you invoke a curious relationship between the regular docecahedron and the cube.

Have fun!

Answer 6

Euclidea has some pretty interesting challenges. To name a few:

• Given two squares, construct a larger square whose area is the sum of the other two.
• Given $\triangle ABC$, construct $\triangle DEF$ such that points $D, E, F$ lie on $AB, BC, AC$ respectively, and the perimeter is minimized.
• Given a point $A$ and parallel lines $m$ and $n$, find points $B$ on $m$ and $C$ on $n$ such that $\triangle ABC$ is equilateral.
• Given a line segment $AB$ and a line parallel to $AB$, partition $AB$ into $n$ equal segments, for any natural $n$. Catch: You may only use a straightedge.
• Given a line $l$ and points $A$ and $B$, which lie on the same side of $l$, construct a circle that passes through $A$ and $B$ and is tangent to $l$.
• Given two points $A$ and $B$, construct the midpoint of $AB$. Catch: You may only use a compass.