We write
C* for the image under f(A,B) of CεH(A,B), and
L* for the image of an arc of a hyperbolic circle through A and B.
result 2
Let L be an arc of a hyperbolic circle through A and B.
If A,B,C,D lie in this order on L, then
A*,B*,C*,D* lie in this order on L* .
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proof
As A,B,C,D
lie in this order on L, ABCD is convex cyclic.
By result 1, C*,D* lie on L*.
We want to show that A*,B*,C*,D* lie in this order on the image.
The alternative is that they lie in order A*,B*,D*,C*, so that
A*B*D*C* is convex. Then, applying the euclidean version of,
ptolemy's theorem to the cyclic euclidean quadrilateral A*B*D*C*,
|A*D*||B*C*| = |A*B*||C*D|+|A*C*|B*D*|>|A*C*||B*D*|.
By the definition of f(A,B),
|A*D*|=s(AD), |B*C*|=s(BC), |A*C*|=s(AC), |B*D*|=s(BD|, so
s(AD)s(BC)>s(AC)s(BD).
But applying the hyperbolic version of ptolemy's theorem to ABCD,
s(AC)s(BD)=s(AB)s(CD)+s(AD)s(BC)>s(AD)s(BC).
This gives a contradiction, so we have the result.
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