heron's formula for hyperbolic triangles If a hyperbolic triangle has sides of lengths a, b and c, then its area D is given by cos(D) = (α+β+γ+αβ+βγ+γα+α2+β2+γ2- αβγ)/(1+α)(1+β)(1+γ), and hence by tan(D/2) = Δ/(1+α+β+γ), where α=cosh(a),β=cosh(b),γ=cosh(c), and Δ2 = 1-α2-β2-γ2+2αβγ. Also, sin(D/2) = Δ/4cosh(½a)cosh(½b)cosh(½c)
proof
The cosine rule for hyperbolic triangles gives
The sine rule for hyperbolic triangles gives
Standard trigonometric addition formula give
If we multiply the last by (sinh(a)sinh(b)sinh(c))2, and substitute for the sines and cosines,
Now sinh2(a) = cosh(a)2-1 = α2-1, so we get
By the gauss-bonnet formula, the area D is given by π-(A+B+C),
The second is derived by using the identity :
tan2(x/2) = (1-cos(x))/(1+cos(x)),
If we use this to derive tan2(D/2) from cos(D) and simplify considerably, we get
The final result follows from the fact that cos(D)+1 = 2sin2(D/2), so that, after tidying,
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