In the course of the proof of the size lemma, we saw that if
a hyperbolic triangle ΔABC has circumcircle of radius r, then
2r = a if and only if cosh(a) + 1 = cosh(b) + cosh(c).
To interpret this, we can use a Corollary of
Stewart's Theorem
Corollary
If AD is a median of the hyperbolic triangle ABC,
then d = d(A,D)
satisfies cosh(d) = (cosh(b)+cosh(c))/2cosh(½a).
Now cosh(a) + 1 = 2cosh2(½a), so that
cosh(a) + 1 = cosh(b) + cosh(c)
if and only if
cosh(½a)= (cosh(b)+cosh(c))/2cosh(½a)
By the Corollary, this is equivalent to the condition
"the median AD has length ½a".
As AD is a median, d(B,D) = d(C,D) = ½a, so that
the condition is "D is the circumcentre".
We can interpret this as:
The vertices of hyperbolic triangle ΔABC lie on a hyperbolic circle
with BC as diameter if and only if cosh(a)+1 = cosh(b)+cosh(c).
If we observe that cosh(2x) = 2sinh2(x)+1, then we arrive at the
familiar looking
The vertices of hyperbolic triangle ΔABC lie on a hyperbolic circle
with BC as diameter if and only if A2 = B2 + C2,
where A = sinh(½a), B = sinh(½b), C = sinh(½c).
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