From baby Hartshorne: showing the exterior of a circle is segment connected

The claim made

... that the exterior of a circle is a segment connected set, which means that if $B$ and $C$ are two points outside a circle $\Gamma$, then there exists a third point $D$ such that $\overline{BD}$ and $\overline{DC}$ are entirely outside $\Gamma$.

is not true in neutral plane geometry (a.k.a. the Hilbert plane). More precisely, it is not provable since it is true in Euclidean geometry and false in hyperbolic geometry, both of which are models of neutral geometry.

However, using the definition of a segment-connected set given in Hartshorne's Geometry page 80:

A subset $W$ of the plane is segment-connected if given any two points $A, B \in W$, there is a finite sequence of points $A = A_1, A_2, \dots , A_n = B$ such that for each $i = 1, 2,..., n-1$, the segment $\overline {A_iA_{i+1}}$ is entirely contained within $W$,

it can be shown that, in neutral geometry, the exterior of a circle is segment-connected.

Proof that the claim is true in Euclidean geometry

Let circle $\Gamma$ have radius $r$ and center $O$. If the line $\ell$ connecting $B$ and $C$ does not pass through $\Gamma$, we are done. Otherwise, let us construct a line $b$ passing through $B$ perpendicular to $\overleftrightarrow{OB}$ and a line $c$ passing through $C$ perpendicular to $\overleftrightarrow{OC}$ If line $\ell$ does not pass through $O$, $B$ and $C$ are on the same side of a line $m$ parallel to $\ell$ and passing through $O$, and the measure of angle $\angle BOC$ is less than $180$. If we consider $\ell$ a transversal cutting lines $b$ and $c$ and consecutive interior angles $\beta$ and $\gamma$ sum to less than $180$. Thus by Euclid's fifth postulate lines $b$ and $c$ intersect at some point $D$ and because all points on $b$ are at least $OB$ from $O$ and all points on $c$ are at least $OC$ from $O$ the segments $\overline{BD}$ and $\overline{DC}$ are entirely outside $\Gamma$.

If $\ell$ does pass through $O$, then without loss of generality let us assume $OB \leq OC$. Set $\epsilon = \frac{OB-r}{2}$ and construct a line $n$ perpendicular to $\ell$ at $O$. Find a point $O'$ on $n$ a distance $\epsilon$ from $O$. Let us now consider a second circle $\Gamma'$ with center $O'$ and radius $r+\epsilon$. The circle $\Gamma'$ fully contains $\Gamma$ since, if point $X$ is in or on $\Gamma$, $OX \leq r$ and, by the triangle inequality, $O'X \leq OX + \epsilon \leq r+ \epsilon$. So $X$ is in or on $\Gamma'$. Point $B$ is not in $\Gamma'$ since $m$ is perpendicular to $\ell$ at $O$ and $O'B > OB = r + 2\epsilon > r+ \epsilon$. We get a similar result for point $C$. Hence, we can apply the first part of this proof to $B$, $C$ and the new circle $\Gamma'$ to address this case.

Proof that the claim is false in hyperbolic geometry

Consider any circle $\Gamma$ with center $O$ and radius $r$ and with diameter $\overline{QQ'}$. Let $\ell$ be the line perpendicular to line $\overleftrightarrow{QQ'}$ at $O$, and let $P$ and $P'$ be the points of $\ell$ which intersect the circle. Let $m$ be the line passing through $P$ that is asymptotic to line $\overleftrightarrow{QQ'}$ on the $Q$ side of $O$. Drop a perpendicular from $O$ to the line $m$, and let $F$ be the foot of that perpendicular. The point $F$ will be the closest approach that $m$ comes to $O$ and, since $m$ is not perpendicular to $\ell$ at $P$, $OF < r$. Let $R$ be a point of $\overrightarrow{OF}$ between $F$ and the circle.

Let $n$ be the line through $R$ that is asymptotic to $m$ in the other direction. since $m$ and $n$ are asymptotic, any line passing through one, in the direction of the asymptote, passes through the other. So the line $\ell$ passes through both. Consider any one of the points of $\ell$ between $m$ and $n$ — call it $B$. Any line through $B$ that avoids the circle must pass through $n$ and must also pass through $m$ but on the $Q'$ side of $\ell$. It consequently cannot cross $m$ on the $Q$ side and so cannot cross the ray $\overrightarrow{OQ}$. We can make the same argument on the other side of $\ell$ to find points, $B$, near $P$ just outside the circle such that if a line passes through $B$ but not the circle, it cannot intersect ray $\overrightarrow{OQ'}$. There is, apparently, no line through these points that can avoid the circle and go across line $\overleftrightarrow{QQ'}$. Of course, we can make a similar argument about points, $C$, near $P'$ just outside the circle. However, if we are to find a $D$ for a pair of these $B$'s and $C$'s such that $\overline{BD}$ and $\overline{DC}$ are entirely outside $\Gamma$, one or the other of $\overline{BD}$ or $\overline{DC}$ must cross $\overleftrightarrow{QQ'}$.

We have shown that in hyperbolic geometry there are points in the exterior of any circle that require more than two segments to connect them and stay in the exterior. Fooling around with the Poincaré disk, reveals that the larger the circle, the more segments may be needed.

Proof that, in neutral geometry, the exterior of a circle is segment-connected

Consider any circle $\Gamma$ with center $O$ and radius $r$ and any points $B$ and $C$ in the circle's exterior. Construct a perpendicular line $m_1$ to line $\overleftrightarrow{OB}$ at point $B$. Note that all of line $m_1$ lies outside the circle. Find a point $A_1$ on line $m_1$ to one side of $B$ at a distance of $2r$ from $B$ and let angle $\theta$ be the measure of angle $\angle BOA_1$. If $C$ falls on or in the interior of angle $\angle BOA_1$, then the ray $\overrightarrow{OC}$ will intersect $m_1$ at a point $E$. Connect $B$ to $E$ with a segment that lies on $m_1$ and so lies entirely outside $\Gamma$, and then finish by connecting $E$ to $C$ with a segment which will also lie outside since $E$ ad $C$ both lie outside $\Gamma$. If $C$ does not fall in the interior of angle $\angle BOA_1$, connect B to $A_1$, and then find the next point of our segment connection on ray $\overrightarrow{OA_1}$ at distance $OB$ from $O$ and call the point $B_1$.

Now repeat the process with $B_1$. Construct a perpendicular line $m_2$ to line $\overleftrightarrow{OB_1}$ at point $B_1$, and find a point $A_2$ on line $m_2$ to the side of $B_1$ opposite $B$ at a distance of $2r$ from $B_1$. By SAS, the triangle $\triangle OBA_1$ is congruent to $\triangle OB_1A_2$ and so the measure of the angle $\angle B_1OA_2$ is also $\theta$. Again, if $C$ falls on or in the interior of angle $\angle B_1OA_2$, then we can find, as before, a point $E$ on $m_2$ and connect $B_1$ to $E$ to $C$ with segments. If not, connect to $A_2$ and find the next point of our segment connection on ray $\overrightarrow{OA_2}$ at distance $OB$ from $O$ and call the point $B_2$.

We can repeat the process until, by Archimedian property of angle measures, $C$ lies within $\angle B_iOA_{i+1}$ by which time we will have constructed a segment connection between $B$ and $C$ entirely in the exterior of $\Gamma$.

hmm... I accidentally stumbled upon this post and I was thinking:

Proof that the claim is true in Euclidean geometry

Let circle Γ have radius r and center O. If the line ℓ connecting B and C does not pass through Γ, we are done. Otherwise, let us construct a line b passing through B perpendicular to OB←→ and a line c passing through C perpendicular to OC←→ If line ℓ does not pass through O, B and C are on the same side of a line m parallel to ℓ and passing through O,

My proof is similar up to here, then I took a different approach.

Assume b // c

Then extend the line OB s.t. it intersects c. Let E be that intersection. Then angle OBG, where G belongs to b, would congruent to angle OEC. Then you have angle OEC and OCE both being right angle.

The contradiction (Greenberg Proposition 4.5 I think...) should yield the point D needed for this proof. Then use Greenberg Propostion 4.6 to show all pts on BD & DC are in the exterior of the ciricle.