Instruct geometer moths so you can learn about their true geometry. I had a space-ship wreck in an unknown world of some kind of moths.  I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and compasses. I asked them to construct a tangent lines to a circle $c$ from a point $P$. (So, thought I, I could learn more about their geometry.)
I include the final result I was allowed to take a photo of:

Now, this is what they did:


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*They drew a white circle $c'$ through $P$ around $C$.

*They joined $C$ and $P$ with a segment

*They chose a point $X$ on $c$.

*They erected a perpendicular to $XC$ at $X$.

*This perpendicular intersected $c'$ at $X'$

*They copied the angle $\alpha=\angle XX'C$ over $CP$ at $C$.

*The red line was apparently a tangent to $c$.


Then they ridiculed me for I could not tell which geometry was their natural one.
Could you order them to construct the tangent so that you be able to tell if they were hyperbolic moths, spherical moths, or Euclidean moths?
I asked the geometer moth's to find a tangent line to another circle from another point $P$. 

In this case I instructed them as follows.


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*Connect $C$ and $P$ with a segment.

*Halve the segment. The middle point was denoted by $H$.

*Construct a circle centered at $H$ through $P$.

*Connect one of the intersection points of  the two circles ($c$, $c'$) with $P$.

*We did not get a tangent line.


The geometer moths confessed that their geometry was not Euclidean. But they did not want to tell if it was hyperbolic or elliptic.
Help needed to make up a method of constructing the tangent line so that one can separate the two remaining geometries.
Don't question the existence of geometer moth's!
 A: It is easier when you ask them to make a square.
Starting with 3 right angles (they are constructable ) what is the measure of the fourth angle? if it is larger than a right angle their geometry is spherical , if it is smaller than a right angle it is hyperbolic
Having a look at your second construction you did get a line that cut the circle in two places. 
The other point where it cut the circle was that between T and P? if so it is hyperbolic.
If T is between P and the second point it is spherical.
the proof sketch is a bit:
Draw the lines PT and PT'  that are  the tangent line to circle c going trough P
In spherical geometry the lines PT, PT' and PC are will eventually meet at the anti pole of P
all lines that intersect circle c between T and T' will have to cut the circle twice . (because they all go trough the antipole of P
All what then remains is to prove that the intersections of circles c and c' are between T and T'
and this can be proven by proving that HT is larger than HC
(and I haven't solved this bit yet)
PS No it is not easier to make a triangle , you get in a mess with very big triangles or very messy measurements, it is much simpler to try squares.
