some theorems of neutral geometry

In neutral geometry, we have lines, segments and circles and the concepts
of length and angle. We can therefore discuss triangles and measure their
interior angles and the lengths of their sides. Since the sum of the angles
of a triangle is at most π, the interior angles have size between 0 and π.

In the first theorem, we consider the size of angles, and assume that each
angle is taken in the range (0,π). That is to say, we take <XYZ as the
measure of the internal angle at vertex Y in ΔXYZ.

the angle-coherence theorem
For any triangle ΔABC, and any point P which is not on any
of the lines AB, BC, CA. Let <APB=γ, <BPC=α, <CPA=β, then
1 + 2cos(α)cos(β)cos(γ) - cos2(α) - cos2(β) - cos2(γ) = 0.

proof

In any triangle, the sum of two sides must exceed the third. This is just the
triangle inequality.

Definition
A triple {a,b,c} of positive numbers is side-coherent if there is a triangle with
sides of length a, b and c.

By considering a line BC of length a, and circles round B of radius c, and round
C of radius b, we see that side-coherence is equivalent to the condition that
the sum of any two of a,b,c is greater than the third. This follows since these
circles must intersect in a point A, and ΔABC is the required triangle.

the side-coherence theorem
The side-coherence of the triple {a,b,c} is equivalent to either of the conditions
(1) Γ2 = (a+b+c)(a+b-c)(b+c-a)(c+a-b) > 0,
(2) Δ2 = 1 + 2cosh(a)cosh(b)cosh(c) - cosh2(a) - cosh2(b) - cosh2(c) > 0.

proof

If we have a side-coherent triple, then we may define Γ and Δ as the positive
roots of the quantities Γ2 and Δ2.

If we are working in euclidean geometry, then (1) appears more natural. Indeed,
if we write s = (a+b+c)/2, then Γ2 =16s(s-a)(s-b)(s-c). Then, by Heron's formula,
Γ is 4 times the euclidean area of the triangle.

In hyperbolic geometry, we have come to expect hyperbolic functions of lengths,
so (2) might be more significant. In fact, we meet the number Δ throughout the
study of hyperbolic geometry. In particular, it is related to the hyperbolic area,
and is at the heart of the proof of the Sine Rule. See, for example

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