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


In Theorem 10.4 of the same book, we prove that all of Euclid's propositions I.1 through I.28 (with the exception of I.1 and I.22) are valid in a Hilbert plane.

In particular, I.28 is true: If a straight line falling on two straight lines makes the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another. This means that $AC$ is parallel to $\ell$, and so $A$ and $C$ are on the same side of $\ell$.


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