I taught a Euclidean geometry class last year, which I thought was a dream. It is a first course in proofs, yet is not axiomatic... something with which I'm still wrestling.
This summer, I'm trying to improve the approach (in my opinion). One important tool I want to introduce early on is the idea of an affine map. Many proofs greatly simplify and some of the surprising results become plain (yes, I'm a party pooper). The catch is that we prefer synthetic proofs to using coordinates. So, here's the problem:
Start with Euclid's axioms and perhaps some others that you deem suitable. Define an affine map of the plane to be a bijection that takes lines to lines (I believe this is equivalent). Note, this means parallel lines are preserved.
Suppose $A,B,C$ are points on a line and $T$ is an affine map.
Harder problem Prove that the directed segments are in proportion: $$AB:AC=T(A)T(B):T(A)T(C)$$
The harder problem quickly implies that ratios of areas are preserved.
Easier problem Prove that if $B$ is between $A$ and $C$, then $T(B)$ is between $T(A)$ and $T(C)$.
Progress This paper is interesting, although I still don't know what a g-reflection is. I expect a much more basic solution.
Once the easier problem is solved, I will have the harder problem as is shown below. Of course, the easier problem has to do with betweenness. We therefore at least need the notion, say $B$ is between $A$ and $C$ if $AB+BC=AC$. Now, we just need that affine maps (as defined above) preserve this notion. I can find no counterexamples.
Affine maps do not preserve circles, so it seems these problems are related to straightedge-only constructions, such as the coffin problem
You are given two parallel segments. Using a straightedge, divide one of them into six equal parts.
Actually, the solution to this problem is one of those things that will become unsurprising if we can get affine maps going.
To our advantage, we have another tool in addition to the straight edge. We can draw parallel lines through points, since affine maps preserve parallel lines. It is then easy to prove that midpoints map to midpoints by construction.
For any segment $BC$, construct a parallelogram $\square ABCD$. Draw a line parallel to the sides adjacent to $BC$ and through the intersection of the diagonals. It will intersect $BC$ at the midpoint $M$. Following the same construction for $T(A),T(B)$ we see that $T(M)$ must be the midpoint of $T(A)$ and $T(B)$.
Once we have midpoints, we can take midpoints of midpoints, etc. until we have all the ticks on our ruler coinciding through the affine map. I have trouble making the jump to third-way points, much less constructible numbers. The key is in solving the easier problem, which will allow some continuity type argument by repeatedly subdividing and choosing which halves.