In proposition 34.1, Hartshorne proves:
In a Hilbert plane, suppose that two equal perpendiculars $AC$, $BD$ stand at the ends of an interval $AB$, and we join $CD$. Then the angles at $C$ and $D$ are equal, and furthermore the line joining the midpoints of $AB$ and $CD$, the midline, is perpendicular to both.
Let $\ell$ be the perpendicular bisector of $AB$. Early on in the proof, Hartshorne claims that $A$ and $C$ lie on the same side of the line $\ell$. While this is patently true in a Euclidean geometry, I don't see how it follows in a neutral geometry.
Here's a copy of the book posted online: http://www.math.unam.mx/javier/Hartshorne.pdf