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!
Best Answer
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).
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.
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.
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.