TITLE: Geometry Construction Problems
QUESTION [13 upvotes]: 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!

REPLY [3 votes]: 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.