In the standard notation, the area of the euclidean triangle ABC can be written as
If we recall our earlier notation for a hyperbolic triangle ABC -
sinh(b)sinh(c)sin(A) = sinh(a)sinh(b)sin(C) = sinh(a)sinh(c)sin(B) = Δ,
where Δ is the positive root of 1 - α2 - β2 - γ2 + 2αβγ.
Thus the expression sinh(x)sinh(y)sin(Z), where x, y are the hyperbolic lengths of the
The quantity Δ also appears when we investigate the hyperbolic length of altitudes.
Th sketch on the right shows a hyperbolic triangle ABC and the altitude AD at the
vertex A. The triangle ABD has a right angle at D, so we an apply the sine formula
to get sin(B) = sinh(hA)/sinh(c), where hA = d(A,D), the length of the altitude.
From the formulae above, Δ = sinh(a)sinh(c)sin(B), so we then obtain
Δ = sinh(a)sinh(hA). Again by the symmetry of Δ in a,b,c we get the result
sinh(a)sinh(hA) = sinh(b)sinh(hB) = sinh(c)sinh(hC) = Δ.
We also have expressions for the length of the altitudes e.g. sinh(hA) = Δ/sinh(a).
Now we have six expressions for Δ, each of which is an analogue of a euclidean
when we try to prove a hyperbolic version of a euclidean result, then
(1) euclidean lengths are replaced by hyperbolic functions of the hyperbolic lengths,
We offer no serious attempt to justify these phenomena. We merely observe that