Hyperbolic Cosines and Sines Theorems for the Triangle Formed by Arcs of Intersecting Semicircles on Euclidean Plane

Abstract

The hyperbolic cosines and sines theorems for the curvilinear triangle bounded by circular arcs of three intersecting circles are formulated and proved by using the general complex calculus. The method is based on a key formula establishing a relationship between exponential function and the cross-ratio. The proofs are carried out on Euclidean plane.

1. Introduction

Classically, the models of the hyperbolic plane are regarded as based on Euclidean geometry. One starts with a piece of a Euclidean plane, a half-plane, or a circular disk and then, in that half-plane or disk the notions of points, lines, distances, angles are defined as things that could be described in terms of Euclidean geometry [1–3]. The model of the hyperbolic plane is the half-plane model. The underlying space of this model is the upper half-plane model 𝐻 in the complex plane 𝐶, defined to be 𝐻={𝑧∈𝐶∶Im(𝑧)>0}. (1) In coordinates (𝑥,𝑦) the line element is defined as 𝑑𝑠²=1𝑦²𝑑𝑥²+𝑑𝑦². (2) The geodesics of this space are semicircles centered on the 𝑥-axis and vertical half-lines. The geometrical properties of the figures on the half-plane are studied by considering quantities invariant under an action of the general Möbius group, which consists of compositions of Möbius transformations and reflections [4]. The curvilinear triangle formed by circular arcs of three intersecting semicircles is one of the principal figures of the upper half-plane model 𝐻. The hyperbolic laws of sines-cosines for that triangle are proved by using properties of the Möbius group and the upper half-plane 𝐻.

In this paper we suggest another way of construction of proofs of the sines-cosines theorems of the Poincaré model. The curvilinear triangle formed by circular arcs is the figure of the Euclidean plane; consequently, on the Euclidean plane we have to find relationships antecedent to the sines-cosines hyperbolic laws. Therefore, first of all, we establish these relationships by making use of axioms of the Euclidean plane, only. Secondly, we proove that these relationships can be formulated as the hyperbolic sine-cosine theorems. For that purpose we refer to the general complex calculus and within its framework establish a relationship between exponential function and the cross-ratio. In this way the hyperbolic trigonometry emerges on Euclidean plane in a natural way.

The paper is organized as follows. In Section 2  a key formula connecting the hyperbolic calculus with the cross-ratio is established. By employing the key formula and the Pythagoras theorem the elements of the right-angled triangle are expressed as functions of the hyperbolic trigonometry. In Section 3 we explore Euclidian properties of the curvilinear triangle bounded by circular arcs of three intersecting semicircles. Two main relationships connecting three intersecting semicircles are established. In Section 4, we prove the theorem of cosines for the curvilinear triangle bounded by the circular arcs. In Section 5, the hyperbolic law of sines and second hyperbolic law of cosines are derived on the basis of the main relationships between these semicircles.

2. Hyperbolic Trigonometry in Euclidean Geometry

2.1. Trigonometry Induced by General Complex Algebra

The simplest generalization of the complex algebra, denominated as General Complex Algebra, is defined by unique generator 𝐞 satisfying the quadratic equation [5] 𝐞²−𝑏₁𝐞+𝑏₀=0, (3) where coefficients 𝑏₀,𝑏₁ are given by real positive numbers and 𝑏²₁−4𝑏₀>0.

The following Euler formula holds true: exp𝐞𝜙=𝑔₀𝜙+𝐞𝑔₁𝜙. (4) Denote by 𝑥₁,𝑥₂∈𝑅 roots of the quadratic equation (3). The Euler formula (4) is decomposed into two independent equations: exp𝑥_𝑘𝜙=𝑔₀𝜙+𝑥_𝑘𝑔₁𝜙,𝑘=1,2, (5) from which one may find explicit formulae for 𝑔-functions. These functions are linear combinations of the exponential functions exp(𝑥_𝑘𝜙), 𝑘=1,2. Geometrical interpretation of the general complex algebra is done in [6].

Form the following ratio exp𝑥₂−𝑥₁𝜙=𝑥₂−𝐷𝜙𝑥₁−𝐷𝜙, (6) where 𝐷𝜙𝑔=−⁰𝜙𝑔₁𝜙. (7) Introduce a pair of variables 𝑋₁,𝑋₂ by 𝑋_𝑘𝜙=𝑥_𝑘−𝐷𝜙,𝑘=1,2. (8) Then, (6) is written as follows: exp𝑋₂−𝑋₁𝜙=𝑋₂𝜙𝑋₁𝜙. (9) This formula we denominate as the key formula and use it in order to introduce hyperbolic trigonometry on Euclidean plane. Notice that the argument of the exponential function is proportional to the difference between the numerator and the denominator, 𝑋₁−𝑋₂=𝑥₁−𝑥₂, and this difference does not depend on the parameter 𝜙.

2.2. Elements of Right-Angled Triangle as Functions of a Hyperbolic Trigonometry

Let ▵𝐴𝐵𝐶 be a right-angled triangle with right angle at 𝐶. Denote the sides by 𝑎,𝑏, the hypotenuse by 𝑐, and the angles opposite to 𝑎, 𝑏, 𝑐 by 𝐴, 𝐵, 𝐶, correspondingly. Traditionally interrelations between angles and sides of a triangle are described by the trigonometry via periodical sine-cosine functions. The periodical functions of the circular angles are defined via the ratios 𝑏sin𝐵=𝑐𝑏,tan𝐵=𝑎. (10) Notice that these two ratios are functions of the same angle 𝐵. On making use of formula (9) we are able to introduce the hyperbolic trigonometry besides the circular trigonometry. Define the following relationships: 𝑐+𝑎().𝑐−𝑎=exp2𝜉 (11) From this equation it follows that 𝑎𝑐(𝜉),𝑎=tanh𝑏=sinh(𝜉). (12) In this way we arrive to the following interrelations between circular and hyperbolic functions: 𝑎cos𝐵=𝑐(𝜉)𝑏=tanh,tan𝐵=𝑎=1(𝜉).sinh (13)

Now let us introduce the following geometrical motion; namely, change position of the point 𝐴 along the line 𝐴𝐶 (see, Figure 1). Then, the sides 𝑏 and 𝑐 will change while the side 𝑎 will not change. Since the length 𝑎 is a constant of this evolution, in agreement with key formula (9) we rewrite (11) as follows: 𝑐+𝑎,𝑐−𝑎=exp2𝑎𝜙 (14) that is, the argument of exponential function is proportional to 𝑎: 𝜉=𝑎𝜙, where 𝜙 is a parameter of the evolution. Formulae in (13) are rewritten as 𝑎cos𝐵=𝑐𝑏=tanh𝑎𝜙,tan𝐵=𝑎=1.sinh𝑎𝜙 (15) From these formulas it is derived that cot𝐵=sinh𝑎𝜙, (16) or, 𝐵tan2=exp−𝑎𝜙. (17)

Figure 1 

Motion of the hypothenuse of the right angled triangle.

Install the triangle ▵𝐴𝐵𝐶 in such a way that the side 𝑏=𝐴𝐶 lies on the line 𝐿, and the side 𝑎=𝐵𝐶 is perpendicular to this line at the point 𝐶

In Figure 1 the line 𝐴𝐵 moves in such a way that cuts line 𝐿 at points 𝐴, 𝐴^′, 𝐴^′′, and so forth. Now, let us recall the problem of parallel lines in geometry [7]. Let the ray 𝐴𝐵 tend to a definite limiting position; in Figure 1 this position is given by line 𝐿₁. As the point 𝐴 moves along line 𝐿 away from point 𝐶 there are two possibilities to consider.(1) In Euclidean geometry, the angle between lines 𝐿₁ and 𝐵𝐶 is equal to right angle.(2) The hypothesis of hyperbolic geometry is that this angle is less than the right angle.

The most fundamental formula of the hyperbolic geometry is the formula connecting the angle of parallelism Π(𝑎) and the length 𝑎 of the perpendicular from the given point to the given line. In order to establish those relationships the concept of horocycles, some circles with center and axis at infinity, was introduced [8]. The great theorem which enables one to introduce the circular functions, sines, and cosines of an angle is that the geometry of shortest lines (horocycles) traced on horosphere is the same as plane Euclidean geometry. The function connecting the angle of parallelism with the distance 𝑎 is given by −𝑎exp𝜅Π(𝑎)=tan2. (18) Now, we introduce the value inverse to 𝜙 by 𝐊=1/𝜙 and rewrite (17) in the form −𝑎exp𝐊𝐵=tan2. (19)

Let the point 𝐴 run away from the point 𝐶 to infinity. The following two cases can be considered.(1)𝜙 tends to zero, and 𝐊 tends to infinity; then the angle 𝐵 will tend to right angle. This is true in Euclidean plane.(2)Suppose that 𝜙, and 𝐊 and the angle 𝐵 go to some limited values, lim_{𝐴𝐶⟶∞}𝐊=𝜅,lim𝐵=Π(𝑎). (20)In this way, we have established connection of formula (17) with the main formula of hyperbolic geometry (19).

2.3. Rotational Motion of a Line Tangent to the Semicircle

The concept of the circular angle in Euclidean plane is intimately related to the figure of a circle and to motion of a point along the circumference. The hyperbolic angle is also related to the circle because of a motion along the circumference coherence with the motion along the hyperbola [9].

Consider semicircle 𝐂 (Figure 2) with end-points and the center on 𝑥-axis. Denote by 𝐵 the center and by 𝐾₁, 𝐾₂ the end-points of the semicircle. Through end-points of the semicircle, 𝐾₁, 𝐾₂ erect the lines parallel to vertical axis, 𝑌-axis. Draw a line tangent to the semicircle at the point 𝐶; this line crosses 𝑥-axis at the point 𝐴 and intersects with the vertical lines at points 𝑃₁ and 𝑃₂. Draw a line parallel to 𝑌-axis from the center 𝐵 which crosses 𝐂 at the top point 𝑁.

Figure 2 

Semicircle and lines tangent to the semicircle.

Denote by 𝑟 radius of the circle, so that 𝑟=𝐵𝑁=𝐵𝐶 and 𝑟=𝐾₁𝐾₂. Denote by 𝐵 the angle ∠𝐴𝐵𝐶. The triangle ▵𝐴𝐵𝐶 is a right triangle, so that (𝐴𝐵)²−(𝐴𝐶)²=𝑟². (21) Consider rotational motion of the line tangent to the semicircle at the point 𝐂. In Figure 2, two positions of this line are given by lines 𝐴^′𝐶₂ and 𝐴𝐶₁. When the point 𝐶runs from end point 𝐾₁ to the top-point 𝑁, the point 𝐴 runs along 𝑥-axis from point 𝐾₁ to infinity. During this motion the triangle formed by the line tangent to the semicircle, the 𝑥-axis and the radius of the circle remains to be right angled. This is exactly the case considered in Section 2.2, consequently, we can apply formula (13). According to (13) we write =𝑟𝐴𝐵=𝑟coth𝑟𝜙cos𝐵,𝐴𝐶₁=𝑟sinh𝑟𝜙=𝑟tan𝐵. (22) Since ▵𝐴𝐵𝐶₁∼▵𝐴𝐾₁𝑃₁, we have 𝑃₁𝐾₁𝐴𝐾₁=𝑟𝐴𝐶₁,𝑃=sinh𝑟𝜙₁𝐾₁=𝐴𝐾₁sinh𝑟𝜙=(𝐴𝐵−𝑟)sinh𝑟𝜙=𝑟exp−𝑟𝜙,𝑃₂𝐾₂=𝑟exp𝑟𝜙. (23)(24) On the basis of obtained formulae the following relationships between circular and hyperbolic trigonometric functions are established: (𝐵)=1sin(𝐵),(𝐵).cosh𝑟𝜙,cot=sinh𝑟𝜙cos=tanh𝑟𝜙 (25) If we make the point 𝐶 tend to the top of semicircle 𝑁, the hyperbolic angle 𝜙 will tend to zero. When the point 𝐶 tends to end-point 𝐾₁, the hyperbolic angle tends to infinity. Thus, the hyperbolic angle is measured from the point 𝑁, the top of the semicircle. Consider two different positions of the tangent line, corresponding to two positions 𝐶₁, 𝐶₂, with hyperbolic angles 𝜙₂ and 𝜙₁. According to key formula (6) we write exp𝑥₂−𝑥₁𝜙₂=𝑥₂−𝐷₂𝑥₁−𝐷₂,exp𝑥₂−𝑥₁𝜙₁=𝑥₂−𝐷₁𝑥₁−𝐷₁. (26) The hyperbolic angle 𝜙₂ corresponds to the arc ⌢𝑁𝐶₂, and the hyperbolic angle 𝜙₁ corresponds to the arc ⌢𝑁𝐶₁. Then we suppose that the difference in the hyperbolic angles 𝜙₂−𝜙₁ will correspond to the arc ⌢𝐶₂𝐶₁. From (26) it follows that exp𝑥₂−𝑥₁𝜙₂−𝜙₁=𝑥₂−𝐷₂𝑥₁−𝐷₂𝑥₁−𝐷₁𝑥₂−𝐷₁,𝐷_𝑘=𝐷𝜙_𝑘,𝑘=1,2. (27) It is seen, in the right-hand side we have the cross-ratio. Now, let recall definition of the distance between two points of the geodesic line in the Poincaré model. Let 𝑧,𝑤∈𝐻, and let 𝑧₁ and 𝑤₁ be end points of a geodesic line passing through 𝑧 and 𝑤. Then the distance between these points is defined by formula 𝜌𝑧₁,𝑧₃=log𝑧₁,𝑧₂;𝑧₃,𝑧₄, (28) where 𝑧₁,𝑧₂;𝑧₃,𝑧₄𝑧∶=²−𝑧₁𝑧₄−𝑧₁𝑧₄−𝑧₃𝑧₂−𝑧₃ (29) is the cross-ratio. Comparing (29) with (27) we come to the following correspondence: 𝑧₂=𝑥₂, 𝑧₄=𝑥₁, 𝑧₁=𝐷₂, 𝑧₃=𝐷₁. Notice, however, in our construction 𝐷_𝑘, 𝑘=1,2 are projections of 𝑧₁, 𝑧₃ on 𝑋-axis. The semicircle 𝐂 is the geodesic line, and 𝑥₁, 𝑥₂ are end-points of the geodesic line.

3. Relationships between Elements of Three Intersecting Semicircles

3.1. Hyperbolic Cosine-Sine Functions of Arcs of the Semicircle

The key formula (6) admits to define hyperbolic trigonometric functions of the arcs originated from the top 𝑁 of the semicircle 𝐂. In order to determine trigonometric functions of the arcs with arbitrary end-points on the semicircle we have to use formula (27). Consider arc ⌢𝐶₁𝐶₂ is defined in the first quadrant of the semicircle with end-points at 𝐶₁ and 𝐶₂. The arc ⌢𝐶₁𝐶₂ can be presented as difference of two arcs, both originated from the top of the semicircle: ⌢𝐶₁𝐶₂=⌢𝑁𝐶₁−⌢𝑁𝐶₂. (30) Denote by 𝑎₁, 𝑎₂ the angles formed by radiuses 𝐵𝐶₁ and 𝐵𝐶₂ with 𝑥-axis, correspondingly, where 𝐵 is a center of the circle. Denote the hyperbolic angle corresponding to the arcs ⌢𝑁𝐶_𝑘, 𝑘=1,2 by 𝜉(𝑁𝐶_𝑘), 𝑘=1,2. Then the functions cosh𝜉𝑁𝐶₁,cosh𝜉𝑁𝐶₂,sinh𝜉𝑁𝐶₁,sinh𝜉𝑁𝐶₂ (31) are expressed via circular trigonometric functions according to formulae (25): cosh𝜉𝑁𝐶₁=1sin𝑎₁,cosh𝜉𝑁𝐶₂=1sin𝑎₂,sinh𝜉𝑁𝐶₁=cot𝑎₁,sinh𝜉𝑁𝐶₂=cot𝑎₂. (32) Then, by taking into account additional formulae cosh𝜉𝐶₁𝐶₂=cosh𝜉𝑁𝐶₁cosh𝜉𝑁𝐶₂−sinh𝜉𝑁𝐶₁sinh𝜉𝑁𝐶₂,sinh𝜉𝐶₁𝐶₂=sinh𝜉𝑁𝐶₁cosh𝜉𝑁𝐶₂−cosh𝜉𝑁𝐶₁sinh𝜉𝑁𝐶₂, (33) we get hyperbolic trigonometric functions of the arcs with arbitrary end-points on the semicircle expressed via periodic trigonometric functions:cosh𝜉𝐶₁𝐶₂=1−cos𝑎₁cos𝑎₂sin𝑎₁sin𝑎₂,sinh𝜉𝐶₁𝐶₂=cos𝑎₁−cos𝑎₂sin𝑎₁sin𝑎₂. (34a) Compare with analogous formula from Poincaré model ([3], formula (1.2.6)) given by ()||cosh𝜌𝑧,𝑤=1+𝑧−𝑤²(𝑧)(𝑤).2ImIm (34b)Two complex numbers 𝑧=𝑣₀+𝑖𝑣,𝑤=𝑤₀+𝑖𝑤 (35) are related to the radius of the semicircle and the angles 𝑎₁,𝑎₂ by 𝑣=𝑟sin𝑎₂,𝑣₀=𝑟cos𝑎₂,𝑤=𝑟sin𝑎₁,𝑤₀=𝑟sin𝑎₁. (36) Then, cosh𝜉𝐶₁𝐶₂=1−cos𝑎₁cos𝑎₂sin𝑎₁sin𝑎₂=𝑟²−𝑣₀𝑤₀𝑣𝑤𝑣=1+⁰−𝑤₀²+(𝑣−𝑤)²().2𝑤𝑣=cosh𝜌𝑧,𝑤 (37)

3.2. Relationships between Elements of the Triangle Bounded by Arcs of the Intersecting Semicircles

In Figure 3 three intersecting semicircles with centers installed on horizontal axis at the points 𝑂_𝑎, 𝑂_𝑏, and 𝑂_𝑐 are presented. Intersections of the semicircles form triangle ▵𝐴𝐵𝐶 bounded by the arcs ⌢𝐵𝐶,⌢𝐴𝐵,⌢𝐴𝐶. For each arc we can put in correspondence the hyperbolic angle. Denote the hyperbolic angles by 𝑎,𝑏,𝑐, where 𝑎, 𝑏, 𝑐 consecutively will correspond to the arcs ⌢𝐵𝐶,⌢𝐴𝐶,⌢𝐴𝐵.

Figure 3 

Triangle formed by arcs of intersecting semicircles.

Connect vertices of the triangle with centers of the circle by corresponding radiuses. Denote by 𝑎_𝑘, 𝑏_𝑘, 𝑐_𝑘, 𝑘=1,2 the angles bounded by the radiuses and 𝑥-axis, where 𝑎₁>𝑎₂>0, 𝑏₁>𝑏₂>0, 𝑐₁>𝑐₂>0. By making use of (34a) define hyperbolic cosine-sine functions corresponding to bounding segments:cosh𝑎=1−cos𝑎₁cos𝑎₂sin𝑎₁sin𝑎₂,sinh𝑎=cos𝑎₁−cos𝑎₂sin𝑎₁sin𝑎₂,cosh𝑏=1−cos𝑏₁cos𝑏₂sin𝑏₁sin𝑏₂,sinh𝑏=cos𝑏₁−cos𝑏₂sin𝑏₁sin𝑏₂,cosh𝑐=1−cos𝑐₁cos𝑐₂sin𝑐₁sin𝑐₂,sinh𝑐=cos𝑐₁−cos𝑐₂sin𝑐₁sin𝑐₂. (38a)(38b)(38c)The usual notion of the angle is used that is, the angle between two curves is defined as an angle between their tangent lines. Let the angles 𝛼,𝛽,𝛿 be angles at the verteices 𝐴,𝐵,𝐶, correspondingly. For these angles we can define thier proper cosine and sine functions. The angles of the triangle ▵𝐴𝐵𝐶𝛼,𝛽,𝛿 are closely related to angles 𝑎₁,𝑎₂,𝑏₁,𝑏₂,𝑐₁,𝑐₂. From Figure 3 we find the following relationships between them: 𝛽=𝑎₂+𝑐₂,𝛿=𝜋−𝑎₁−𝑏₂,𝛼=𝑏₁−𝑐₁. (39) Then,cos𝛼=cos𝑏₁cos𝑐₁+sin𝑏₁sin𝑐₁,cos𝛽=cos𝑎₂cos𝑐₂−sin𝑎₂sin𝑐₂,cos𝛿=−cos𝑏₂cos𝑎₁+sin𝑏₂sin𝑎₁, (40a)(40b)(40c)sin𝛼=sin𝑏₁cos𝑐₁−cos𝑏₁sin𝑐₁,sin𝛽=sin𝑎₂cos𝑐₂+cos𝑎₂sin𝑐₂,sin𝛿=sin𝑏₂cos𝑎₁+cos𝑏₂sin𝑎₁. (41a)(41b)(41c)

Denote distances between centers by 𝑂_𝑐𝑏=𝑂_𝑐𝑂_𝑏,𝑂_𝑏𝑎=𝑂_𝑏𝑂_𝑎,𝑂_𝑎𝑐=𝑂_𝑎𝑂_𝑐. (42) The theorem of sines employed for triangles 𝑂_𝑐𝐴𝑂_𝑏, 𝑂_𝑏𝐶𝑂_𝑎, 𝑂_𝑐𝐵𝑂_𝑎 gives six relations of typesin𝛼𝑂_𝑐𝑏=sin𝑐₁𝑟_𝑏=sin𝑏₁𝑟_𝑐,sin𝛿𝑂_𝑏𝑎=sin𝑎₁𝑟_𝑏=sin𝑏₂𝑟_𝑎,sin𝛽𝑂_𝑎𝑐=sin𝑐₂𝑟_𝑎=sin𝑎₂𝑟_𝑐. (43a)(43b)(43c)

From these relations the first set of main relationships follows.

Relation 1. Consider 𝑟_𝑎sin𝑎₁=𝑟_𝑏sin𝑏₂,𝑟_𝑐sin𝑐₂=𝑟_𝑎sin𝑎₂,𝑟_𝑐sin𝑐₁=𝑟_𝑏sin𝑏₁. (44)From the draught in Figure 3 it is seen that 𝑂_𝑎𝑐=𝑂_𝑏𝑎+𝑂_𝑐𝑏, (45) where 𝑂_𝑎𝑐=𝑟_𝑐cos𝑐₂+𝑟_𝑎cos𝑎₂,𝑂_𝑏𝑎=𝑟_𝑎cos𝑎₁+𝑟_𝑏cos𝑏₂. (46) Hence, 𝑂_𝑐𝑏=𝑟_𝑐cos𝑐₂+𝑟_𝑎cos𝑎₂−𝑟_𝑎cos𝑎₁−𝑟_𝑏cos𝑏₂. (47) From vertices of ▵𝐴𝐵𝐶 erect lines perpendicular to horizontal line intersect with 𝑥-axis at points ℎ_𝐴,ℎ_𝐵,ℎ_𝐶, correspondingly. From Figure 3 we find that 𝑂_𝑐𝑏=𝑂_𝑐ℎ_𝐴−ℎ_𝐴𝑂_𝑏=𝑟_𝑐cos𝑐₁−𝑟_𝑏cos𝑏₁. (48) By equating (47) with (48) we arrive to another main relationship between radii and angles.

Relation 2. One has 𝑟_𝑐cos𝑐₁−𝑟_𝑏cos𝑏₁=𝑟_𝑐cos𝑐₂+𝑟_𝑎cos𝑎₂−𝑟_𝑎cos𝑎₁−𝑟_𝑏cos𝑏₂. (49)
We will effect a simplification by using the following designations: 𝑟_𝑎cos𝑎₁=𝑤₀₁,𝑟_𝑎sin𝑎₁=𝑤₁,𝑟_𝑎cos𝑎₂=𝑣₀₁,𝑟_𝑎sin𝑎₂=𝑣₁,𝑟_𝑏cos𝑏₁=𝑤₀₂,𝑟_𝑏sin𝑏₁=𝑤₂,𝑟_𝑏cos𝑏₂=𝑣₀₂,𝑟_𝑏sin𝑏₂=𝑣₂,𝑟_𝑐cos𝑐₁=𝑤₀₃,𝑟_𝑐sin𝑐₁=𝑤₃,𝑟_𝑐cos𝑐₂=𝑣₀₃,𝑟_𝑐sin𝑐₂=𝑣₃. (50) In these designations formulae (40a), (40b), (40c) and (41a), (41b), (41c) are written as follows:𝑟_𝑏𝑟_𝑐sin𝛼=𝑤₂𝑤₀₃−𝑤₀₂𝑤₃,𝑟_𝑏𝑟_𝑐cos𝛼=𝑤₀₂𝑤₀₃+𝑤₂𝑤₃,𝑟_𝑎𝑟_𝑐sin𝛽=𝑣₁𝑣₀₃+𝑣₀₁𝑣₃,𝑟_𝑎𝑟_𝑐cos𝛽=𝑣₀₁𝑣₀₃−𝑣₁𝑣₃,𝑟_𝑎𝑟_𝑏sin𝛿=𝑣₂𝑤₀₁+𝑣₀₂𝑤₁,𝑟_𝑎𝑟_𝑏cos𝛿=𝑣₂𝑤₁−𝑣₀₂𝑤₀₁. (51a)(51b)(51c)Formulae (38a), (38b), (38c) for hyperbolic sines and cosines are rewritten as follows: 𝑟cosh𝑎=^2𝑎−𝑤₀₁𝑣₀₁𝑤₁𝑣₁,sinh𝑎=𝑟_𝑎𝑤₀₁−𝑣₀₁𝑤₁𝑣₁,𝑟cosh𝑏=^2𝑏−𝑤₀₂𝑣₀₂𝑤₂𝑣₂,sinh𝑏=𝑟_𝑏𝑤₀₂−𝑣₀₂𝑤₂𝑣₂,𝑟cosh𝑐=^2𝑐−𝑤₀₃𝑣₀₃𝑤₃𝑣₃,sinh𝑐=𝑟_𝑐𝑤₀₃−𝑣₀₃𝑤₃𝑣₃. (52) In the designations equations of Relation 1 are written as 𝑥∶=𝑤₂=𝑤₃,𝑦∶=𝑣₁=𝑣₃,𝑧∶=𝑤₁=𝑣₂. (53) Equations (43a), (43b), (43c)–(46) are rewritten as follows: 𝑂_𝑐𝑏=𝑤₀₃−𝑤₀₂,𝑂_𝑎𝑐=𝑣₀₃+𝑣₀₁,𝑂_𝑏𝑎=𝑤₀₁+𝑣₀₂. (54) Correspondingly, Relation 2 takes the form 𝑤₀₃−𝑤₀₂=𝑣₀₃+𝑣₀₁−𝑤₀₁−𝑣₀₂. (55) This expresses the fact that 𝑂_𝑐𝑎 is a sum of 𝑂_𝑐𝑏 and 𝑂_𝑏𝑎. Notice that (55) can be rewritten also in another equivalent form namely, 𝑤₀₃−𝑣₀₃=𝑤₀₂−𝑣₀₂−𝑤₀₁−𝑣₀₁. (56) Denote the segments projections of sides of ▵𝐴𝐵𝐶 on 𝑥-axis by 𝐏(𝐴𝐶)=ℎ_𝐴ℎ_𝐶, 𝐏(𝐴𝐵)=ℎ_𝐴ℎ_𝐵, 𝐏(𝐵𝐶)=ℎ_𝐵ℎ_𝐶. From Figure 3 it is seen that 𝐏(𝐴𝐶)=𝐏(𝐴𝐵)+𝐏(𝐵𝐶), (57) where 𝐏(𝐵𝐶)=𝑤₀₁−𝑣₀₁,𝐏(𝐴𝐶)=𝑤₀₂−𝑣₀₂,𝐏(𝐴𝐵)=𝑤₀₃−𝑣₀₃. (58)

4. Hyperbolic Law of Cosines I for the Triangle Formed by Intersection of Three Semicircles

The main aim of this section is to prove the hyperbolic law theorem of cosines I for the triangle ▵𝐴𝐵𝐶 formed by arcs of intersecting semicircles with centers installed on 𝑥-axis (Figure 3). This law is given by the following set of equations: cosh𝑐=cosh𝑎cosh𝑏−sinh𝑎sinh𝑏cos𝛿,cosh𝑏=cosh𝑎cosh𝑐−sinh𝑐sinh𝑎cos𝛽,cosh𝑎=cosh𝑐cosh𝑏−sinh𝑐sinh𝑏cos𝛼. (59)

Theorem 3 (theorem of cosines I). The following equation for elements of the triangle ▵𝐴𝐵𝐶 formed by arcs of three intersecting semicircles holds true cosh𝑐=cosh𝑎cosh𝑏−sinh𝑎sinh𝑏cos𝛿, (60) where cosh𝑐,cosh𝑎,cosh𝑏,sinh𝑎,sinh𝑏,cos𝛿 are defined by formulae (51a), (51b), (51c)-(52).

Proof. Square both sides of Relation 2 to obtain 𝑤₀₃−𝑣₀₃²=𝑤₀₂−𝑣₀₂²+𝑤₀₁−𝑣₀₁²−2𝑤₀₂−𝑣₀₂𝑤₀₁−𝑣₀₁ (61) and evaluate this equality by taking into account formulae (51a), (51b), (51c)-(52). First of all, evaluate the left-hand side of this equation as follows: 𝑤₀₃−𝑣₀₃²=𝑣²₀₃+𝑤²₀₃−2𝑣₀₃𝑤₀₃=𝑣²₀₃+𝑤²₀₃−2𝑟²_𝑐+2𝑟²_𝑐−𝑣₀₃𝑤₀₃, (62) where 2𝑟²_𝑐=𝑤²₀₃+𝑤²₃+𝑣²₀₃+𝑣²₃. (63) Use (63) and write (62) as follows: 𝑣²₀₃+𝑤²₀₃−2𝑟²_𝑐=𝑣²₀₃+𝑤²₀₃−𝑤²₀₃+𝑤²₃+𝑣²₀₃+𝑣²₃=−𝑤²₃+𝑣²₃. (64) Equation (61) takes the following form: −𝑤²₃+𝑣²₃+2𝑟²_𝑐−𝑣₀₃𝑤₀₃=𝑤₀₂−𝑣₀₂²+𝑤₀₁−𝑣₀₁²−2𝑤₀₂−𝑣₀₂𝑤₀₁−𝑣₀₁. (65) The underlined term passes to the right-hand side of the equation. Then in the left-hand side remains the expression 2𝑟²_𝑐−𝑣₀₃𝑤₀₃=2𝑣₃𝑤₃cosh𝑐. (66) Dividing both sides of the obtained equation by 𝑣₃𝑤₃, we get 12cosh𝑐=𝑣₃𝑤₃𝑤²₃+𝑣²₃+𝑣₀₂−𝑤₀₂²+𝑣₀₁−𝑤₀₁²−2𝑣₀₂−𝑤₀₂𝑣₀₁−𝑤₀₁, (67) according to Relation 1  𝑤₃=𝑤₂, 𝑣₃=𝑣₁. These relations make true the following equation 1𝑤₃𝑣₃=1𝑤₂𝑣₂1𝑤₁𝑣₁𝑣₂𝑤₁. (68) On making use this formula in the right-hand side of (67) we come to the following equation: 12cosh𝑐=𝑤₂𝑣₂1𝑤₁𝑣₁×𝑣₂𝑤₁𝑤²₃+𝑣²₃+𝑣₀₂−𝑤₀₂²+𝑣₀₁−𝑤₀₁²1−2𝑤₂𝑣₂1𝑤₁𝑣₁𝑣₂𝑤₁𝑣₀₂−𝑤₀₂𝑣₀₁−𝑤₀₁. (69) Evaluate now the underlined term, which we firstly write as follows: 2𝑣₀₂−𝑤₀₂𝑣₀₁−𝑤₀₁1𝑤₂𝑣₂1𝑤₁𝑣₁𝑣₂𝑤₁𝑣=2²𝑤₁𝑟_𝑎𝑟_𝑏𝑟_𝑏𝑣₀₂−𝑤₀₂𝑤₂𝑣₂𝑟_𝑎𝑣₀₁−𝑤₀₁𝑤₁𝑣₁. (70) From the second equation of (51c) we have 𝑣₂𝑤₁𝑟_𝑎𝑟_𝑏𝑣=cos𝛿+⁰²𝑤₀₁𝑟_𝑎𝑟_𝑏. (71) By making use of (71) in (70), evaluate (70) as follows: 2𝑣₂𝑤₁𝑟_𝑎𝑟_𝑏𝑟_𝑏𝑣₀₂−𝑤₀₂𝑤₂𝑣₂𝑟_𝑎𝑣₀₁−𝑤₀₁𝑤₁𝑣₁𝑣=2cos𝛿+⁰²𝑤₀₁𝑟_𝑎𝑟_𝑏𝑟_𝑏𝑣₀₂−𝑤₀₂𝑤₂𝑣₂𝑟_𝑎𝑣₀₁−𝑤₀₁𝑤₁𝑣₁𝑟=2cos𝛿^𝑏𝑣₀₂−𝑤₀₂𝑤₂𝑣₂𝑟_𝑎𝑣₀₁−𝑤₀₁𝑤₁𝑣₁−2𝑣₀₂𝑤₀₁𝑣₀₂−𝑤₀₂𝑤₂𝑣₂𝑣₀₁−𝑤₀₁𝑤₁𝑣₁=2cos𝛿sinh𝑎sinh𝑏−2𝑣₀₂𝑤₀₁𝑣₀₂−𝑤₀₂𝑤₂𝑣₂𝑣₀₁−𝑤₀₁𝑤₁𝑣₁. (72) Replace the underlined term of (69) by (72), and pass expression containing sines and cosines to the left-hand side of the obtained equation. As a result, we come to the following equation: =12cosh𝑐−2cos𝛿sinh𝑏sinh𝑎𝑤₂𝑣₂𝑤₁𝑣₁𝑣₂𝑤₁𝑤²₂+𝑣²₁+𝑤₀₂−𝑣₀₂²+𝑤₀₁−𝑣₀₁²−2𝑣₀₂𝑤₀₁×𝑤₀₂−𝑣₀₂𝑤₀₁−𝑣₀₁ (73) By applying the elementary algebra one may show that (see, [10]) 𝑣₂𝑤₁𝑤²₂+𝑣²₁+𝑤₀₂−𝑣₀₂²+𝑤₀₁−𝑣₀₁²−2𝑣₀₂𝑤₀₁𝑤₀₂−𝑣₀₂𝑤₀₁−𝑣₀₁=2𝑟²_𝑎−𝑣₀₁𝑤₀₁𝑟²_𝑏−𝑣₀₂𝑤₀₂. (74) Thus, in the right-hand side of (73) we have 21𝑤₁𝑣₁𝑣₂𝑤₂𝑟²_𝑎−𝑣₀₁𝑤₀₁𝑟²_𝑏−𝑣₀₂𝑤₀₂, (75) which according to formulae (52) is nothing else than 2cosh𝑎cosh𝑏. (76) By using it in (73) we arrive to the following equation: 2cosh𝑐−2cosh𝑎cosh𝑏=2cos𝛿sinh𝑎sinh𝑏. (77)

5. Hyperbolic Laws of Sines and Cosines II

The main task of this section is to prove hyperbolic law (theorem) of sines, which is given by the formulae sinh𝑎=sin𝛼sinh𝑏=sin𝛽sinh𝑐,sin𝛿 (78) and the hyperbolic law (theorem) of cosines II given by the formulae cos𝛿=−cos𝛼cos𝛽−sin𝛼sin𝛽cosh𝑐,cos𝛽=−cos𝛼cos𝛿−sin𝛼sin𝛿cosh𝑏,cos𝛼=−cos𝛽cos𝛿−sin𝛿sin𝛽cosh𝑎. (79)(80)(81)

5.1. Hyperbolic Theorem of Sines and Its Geometrical Interpretation on Euclidean Plane

Theorem 4. The ratios of projections of the sides of triangle ▵𝐴𝐵𝐶 on 𝑥-axis to corresponding distances between centers of the semicircles are equal to each other.

Proof. Projections of the sides of ▵𝐴𝐵𝐶 are given by formulae 𝐏_𝑏𝑐=𝑟_𝑎cos𝑎₁−𝑟_𝑎cos𝑎₂,𝐏_𝑐𝑎=𝑟_𝑏cos𝑏₁−𝑟_𝑏cos𝑏₂,𝐏_𝑎𝑏=𝑟_𝑐cos𝑐₁−𝑟_𝑐cos𝑐₂. (82) Distances between centers of the circles have been defined as (see, (46) and (47)) 𝑂_𝑐𝑎=𝑟_𝑐cos𝑐₂+𝑟_𝑎cos𝑎₂,𝑂_𝑏𝑎=𝑟_𝑎cos𝑎₁+𝑟_𝑏cos𝑏₂,𝑂_𝑐𝑏=𝑟_𝑐cos𝑐₁−𝑟_𝑏cos𝑏₁,𝐏_𝑐𝑎=𝐏_𝑏𝑐+𝐏_𝑎𝑏,𝑂_𝑐𝑎=𝑂_𝑏𝑐+𝑂_𝑎𝑏. (83)(84) Relation 1 given by the set of equations 𝑟_𝑎sin𝑎₁=𝑟_𝑏sin𝑏₂,𝑟_𝑐sin𝑐₂=𝑟_𝑎sin𝑎₂,𝑟_𝑐sin𝑐₁=𝑟_𝑏sin𝑏₁ (85) raises to the second power, 𝑟²_𝑎−𝑟²_𝑎cos²𝑎₁=𝑟²_𝑏−𝑟²_𝑏cos²𝑏₂,𝑟²_𝑐−𝑟²_𝑐cos²𝑐₂=𝑟²_𝑎−𝑟²_𝑎cos²𝑎₂,𝑟²_𝑐−𝑟²_𝑐cos²𝑐₁=𝑟²_𝑏−𝑟²_𝑏cos²𝑏₁. (86) Then, square distances 𝑂²_𝑖𝑘,𝑖,𝑘=𝑎,𝑏,𝑐 and use (86). We get 𝑂²_𝑐𝑎=𝑟²_𝑐cos²𝑐₂+𝑟²_𝑎cos²𝑎₂+2𝑟_𝑐𝑟_𝑎cos𝑐₂cos𝑎₂,𝑂²_𝑏𝑎=𝑟²_𝑎cos²𝑎₁+𝑟²_𝑏cos²𝑏₂+2𝑟_𝑎𝑟_𝑏cos𝑎₁cos𝑏₂,𝑂²_𝑐𝑏=𝑟²_𝑐cos²𝑐₁+𝑟²_𝑏cos²𝑏₁−2𝑟_𝑐𝑟_𝑏cos𝑐₁cos𝑏₁. (87)Combine (86) with (87); in this way we come to the following system of equations:(𝑎)𝑂²_𝑐𝑎=𝑟²_𝑐−𝑟²_𝑎+2𝑟_𝑎cos𝑎₂𝑂_𝑎𝑐,(𝑏)𝑂²_𝑐𝑎=𝑟²_𝑎−𝑟²_𝑐+2𝑟_𝑐cos𝑐₂𝑂_𝑎𝑐,(𝑎)𝑂²_𝑏𝑎=𝑟²_𝑎−𝑟²_𝑏+2𝑟_𝑏cos𝑏₂𝑂_𝑏𝑎,(𝑏)𝑂²_𝑏𝑎=𝑟²_𝑏−𝑟²_𝑎+2𝑟_𝑎cos𝑎₁𝑂_𝑏𝑎,(𝑎)𝑂²_𝑐𝑏=𝑟²_𝑏−𝑟²_𝑐+2𝑟_𝑐cos𝑐₁𝑂_𝑐𝑏,(𝑏)𝑂²_𝑐𝑏=𝑟²_𝑐−𝑟²_𝑏−2𝑟_𝑏cos𝑏₁𝑂_𝑐𝑏. (88a)(88b)(88c)From these equations the cosines of the angles 𝑎₁,𝑎₂,𝑏₁,𝑏₂,𝑐₁,𝑐₂ are expressed: 𝑂²_𝑏𝑎−𝑟²_𝑏+𝑟²_𝑎2𝑟_𝑎𝑂_𝑏𝑎=cos𝑎₁,𝑂²_𝑐𝑎−𝑟²_𝑐+𝑟²_𝑎2𝑟_𝑎𝑂_𝑎𝑐=cos𝑎₂,𝑂²_𝑐𝑏−𝑟²_𝑏+𝑟²_𝑐2𝑟_𝑐𝑂_𝑐𝑏=cos𝑐₁,𝑂²_𝑐𝑎−𝑟²_𝑎+𝑟²_𝑐2𝑟_𝑐𝑂_𝑎𝑐=cos𝑐₂,−𝑂²_𝑐𝑏+𝑟²_𝑐−𝑟²_𝑏2𝑟_𝑏𝑂_𝑐𝑏=cos𝑏₁,𝑂²_𝑏𝑎−𝑟²_𝑎+𝑟²_𝑏2𝑟_𝑏𝑂_𝑏𝑎=cos𝑏₂. (89) Having these formulae we may present the projection 𝐏_𝑐𝑎 as follows 𝐏_𝑎𝑐=𝑟_𝑏cos𝑏₁−𝑟_𝑏cos𝑏₂=−𝑂²_𝑐𝑏+𝑟²_𝑐−𝑟²_𝑏2𝑂_𝑐𝑏−𝑂²_𝑏𝑎−𝑟²_𝑎+𝑟²_𝑏2𝑂_𝑏𝑎=−𝑂²_𝑐𝑏𝑂_𝑏𝑎+𝑟²_𝑐𝑂_𝑏𝑎−𝑟²_𝑏𝑂_𝑏𝑎−𝑂²_𝑏𝑎𝑂_𝑐𝑏+𝑟²_𝑎𝑂_𝑐𝑏−𝑟²_𝑏𝑂_𝑐𝑏2𝑂_𝑐𝑏𝑂_𝑏𝑎 (90) The first ratio is presented as follows: 𝐏_𝑐𝑎𝑂_𝑐𝑎=−𝑂_𝑐𝑏𝑂_𝑏𝑎𝑂_𝑐𝑏+𝑂_𝑏𝑎+𝑟²_𝑐𝑂_𝑏𝑎−𝑟²_𝑏𝑂_𝑏𝑎−𝑂_𝑐𝑏+𝑟²_𝑎𝑂_𝑐𝑏2𝑂_𝑐𝑏𝑂_𝑏𝑎𝑂_𝑐𝑎=−𝑂_𝑐𝑏𝑂_𝑏𝑎𝑂_𝑐𝑎+𝑟²_𝑐𝑂_𝑏𝑎−𝑟²_𝑏𝑂_𝑐𝑎+𝑟²_𝑎𝑂_𝑐𝑏2𝑂_𝑐𝑏𝑂_𝑏𝑎𝑂_𝑐𝑎. (91) Now, in the same way let us calculate the next ratio, namely, 𝐏_𝑏𝑐/𝑂_𝑏𝑐. Formula for the projection is evaluated as follows: 𝐏_𝑏𝑐=𝑟_𝑎cos𝑎₁−𝑟_𝑎cos𝑎₂=𝑂²_𝑏𝑎−𝑟²_𝑏+𝑟²_𝑎2𝑂_𝑏𝑎−𝑂²_𝑐𝑎−𝑟²_𝑐+𝑟²_𝑎2𝑂_𝑎𝑐=𝑂_𝑏𝑎𝑂_𝑎𝑐𝑂_𝑏𝑎−𝑂_𝑐𝑎−𝑟²_𝑏𝑂_𝑎𝑐+𝑟²_𝑎𝑂_𝑎𝑐+𝑟²_𝑐𝑂_𝑏𝑎−𝑟²_𝑎𝑂_𝑏𝑎2𝑂_𝑎𝑐𝑂_𝑏𝑎. (92) By taking into account equation −𝑂_𝑐𝑏=𝑂_𝑏𝑎−𝑂_𝑎𝑐, we get 𝐏_𝑏𝑐=𝑂_𝑏𝑎𝑂_𝑎𝑐𝑂_𝑐𝑏−𝑟²_𝑏𝑂_𝑎𝑐+𝑟²_𝑎𝑂_𝑐𝑏+𝑟²_𝑐𝑂_𝑏𝑎2𝑂_𝑎𝑐𝑂_𝑏𝑎. (93) Next, let us calculate the ratio 𝐏_𝑏𝑐𝑂_𝑏𝑐=𝑂_𝑏𝑎𝑂_𝑎𝑐𝑂_𝑐𝑏−𝑟²_𝑏𝑂_𝑎𝑐+𝑟²_𝑎𝑂_𝑐𝑏+𝑟²_𝑐𝑂_𝑏𝑎2𝑂_𝑎𝑐𝑂_𝑏𝑎𝑂_𝑏𝑐. (94) This expression coincides with (91); hence, 𝐏_𝑏𝑐𝑂_𝑏𝑐=𝐏_𝑎𝑐𝑂_𝑎𝑐. (95) By taking into account (84), we arrive to desired relations 𝐏_𝑏𝑐𝑂_𝑏𝑐=𝐏_𝑎𝑐𝑂_𝑎𝑐=𝐏_𝑎𝑏𝑂_𝑎𝑏. (96)

In the sequel come back to designations introduced in Section 3. In these designations (96) is written as follows: 𝑤₀₁−𝑣₀₁𝑤₀₃−𝑤₀₂=𝑤₀₂−𝑣₀₂𝑣₀₃+𝑣₀₁=𝑤₀₃−𝑣₀₃𝑤₀₁+𝑣₀₂. (97)

Theorem 5. The sides and the angles of triangle ▵𝐴𝐵𝐶 satisfy (78).

Proof. By using formulas (38a), (38b), and (38c) for sinh 𝑎, sinh 𝑏, sinh 𝑐 and formulas (41a), (41b), and (41c) for sin 𝛼, sin𝛽,sin𝛿, we write: sinh𝑎=𝑟sin𝛼^𝑎𝑤₀₁−𝑣₀₁∶𝑥𝑤𝑦𝑧⁰³−𝑤₀₂𝑟_𝑏𝑟_𝑐=𝑤₀₁−𝑣₀₁𝑤₀₃−𝑤₀₂𝑥𝑦𝑧𝑟_𝑎𝑟_𝑏𝑟_𝑐,sinh𝑏=𝑤sin𝛽⁰²−𝑣₀₂𝑟_𝑏∶𝑦𝑣𝑥𝑧⁰³+𝑣₀₁𝑟_𝑎𝑟_𝑐=𝑤₀₂−𝑣₀₂𝑣₀₃+𝑣₀₁𝑥𝑦𝑧𝑟_𝑎𝑟_𝑏𝑟_𝑐,sinh𝑐=𝑤sin𝛿⁰³−𝑣₀₃𝑟_𝑐∶𝑧𝑤𝑦𝑥⁰¹+𝑣₀₂𝑟_𝑏𝑟_𝑎=𝑤₀₃−𝑣₀₃𝑤₀₁+𝑣₀₂𝑥𝑦𝑧𝑟_𝑎𝑟_𝑏𝑟_𝑐. (98) It is seen that these equations contain a common factor which is symmetric with respect to 𝑎,𝑏,𝑐 and 𝑥,𝑦,𝑧. Multiply all equations of (97) by this factor. We arrive to (78).

Theorem 6. The sides and the angles of triangle ▵𝐴𝐵𝐶 satisfy the following equation: cos𝛿=sin𝛼sin𝛽cosh𝑐−cos𝛼cos𝛽. (99)

Proof. Evaluate the first term of the right-hand side of (99) by making use of (43a), (43b), and (43c): 1sin𝛼sin𝛽cosh𝑐=𝑟_𝑎𝑟_𝑏𝑂_𝑐𝑎𝑂_𝑐𝑏1−cos𝑐₁cos𝑐₂=1𝑟_𝑎𝑟_𝑏𝑂_𝑐𝑎𝑂_𝑐𝑏×𝑂1−^2𝑐𝑏−𝑟²_𝑏+𝑟²_𝑐2𝑟_𝑐𝑂_𝑐𝑏𝑂²_𝑐𝑎−𝑟²_𝑎+𝑟²_𝑐2𝑟_𝑐𝑂_𝑎𝑐=1𝑟_𝑎𝑟_𝑏𝑂_𝑎𝑐𝑂_𝑐𝑏−1𝑟_𝑎𝑟_𝑏𝑂_𝑎𝑐𝑂_𝑐𝑏×𝑂²_𝑐𝑏−𝑟²_𝑏+𝑟²_𝑐2𝑟_𝑐𝑂_𝑐𝑏𝑂²_𝑎𝑐−𝑟²_𝑎+𝑟²_𝑐2𝑟_𝑐𝑂_𝑎𝑐=14𝑟_𝑎𝑟_𝑏𝑟²_𝑐4𝑂_𝑎𝑐𝑂_𝑐𝑏𝑟²_𝑐−𝑂²_𝑐𝑏𝑂²_𝑐𝑎−𝑂²_𝑐𝑏+𝑂²_𝑐𝑎𝑟²_𝑐+𝑂²_𝑐𝑏𝑟²_𝑎+𝑂²_𝑎𝑐𝑟²_𝑏. (100) Now calculate the product cos𝛼cos𝛽 by using the following formulae: 1cos𝛼=2𝑟_𝑏𝑟_𝑐𝑟²_𝑐+𝑟²_𝑏−𝑂²_𝑐𝑏,1cos𝛽=−2𝑟_𝑎𝑟_𝑐𝑟²_𝑐+𝑟²_𝑎−𝑂²_𝑎𝑐. (101) We obtain 1cos𝛼cos𝛽=2𝑟_𝑏𝑟_𝑐𝑟²_𝑐+𝑟²_𝑏−𝑂²_𝑐𝑏12𝑟_𝑎𝑟_𝑐𝑟²_𝑐+𝑟²_𝑎−𝑂²_𝑎𝑐=14𝑟_𝑏𝑟_𝑎𝑟²_𝑐×𝑂²_𝑐𝑏𝑂²_𝑎𝑐−𝑂²_𝑐𝑏+𝑂²_𝑎𝑐𝑟²_𝑐−𝑂²_𝑐𝑏𝑟²_𝑎−𝑂²_𝑎𝑐𝑟²_𝑏. (102) By using (100) and (102) calculate the difference =1sin𝛼sin𝛽cosh𝑐−cos𝛼cos𝛽4𝑟_𝑎𝑟_𝑏𝑟²_𝑐4𝑂_𝑎𝑐𝑂_𝑐𝑏𝑟²_𝑐−𝑂²_𝑐𝑏𝑂²_𝑎𝑐−𝑂²_𝑐𝑏+𝑂²_𝑎𝑐𝑟²_𝑐+𝑂²_𝑐𝑏𝑟²_𝑎+𝑂²_𝑎𝑐𝑟²_𝑏−14𝑟_𝑏𝑟_𝑎𝑟²_𝑐×+𝑂²_𝑐𝑏𝑂²_𝑎𝑐−𝑂²_𝑐𝑏+𝑂²_𝑎𝑐𝑟²_𝑐−𝑂²_𝑐𝑏𝑟²_𝑎−𝑂²_𝑎𝑐𝑟²_𝑏=12𝑟_𝑏𝑟_𝑎𝑟²_𝑎+𝑟²_𝑏−𝑂²_𝑎𝑏=cos𝛿. (103) Thus, we got (99).

The other two equations, (80) and (81), are proved analogously.

6. Concluding Remarks

We have proved three hyperbolic cosine-sine theorems for the triangular formed by arcs of three intersecting semicircles by using only elements of the Euclidean geometry. This geometrical figure is one of the basic figures of the Poincaré model of hyperbolic geometry. Conventionally, in the textbooks on the Poincaré model of hyperbolic geometry these theorems are proved by employing invariance properties of the cross-ratio with respect to Möbius group transformations. In that case one remains with the impression that the hyperbolic cosine-sine theorems are consequences of a special structure of the 𝐻 half-plane. We have shown that the hyperbolic trigonometry as well as periodic trigonometry arises on Euclidean plane in a natural way.

The method of parametrization of the mass-shell equation via the hyperbolic angle, the rapidity, is widely used in the relativistic physics [4]. The quadratic equation (3) is closely related to the mass-shell equation 𝑝²₀−𝑝²=𝑚²𝑐², (104) where 𝑚𝑐 plays the role of radius of the semicircle 2𝑚𝑐=𝑥₂−𝑥₁. Within the framework of the present method we come to the following parametrization of energy momentum of the relativistic particle [11]: 𝑝₀=𝑚𝑐coth𝑚𝑐𝜙,𝑝=𝑚𝑐sinh𝑚𝑐𝜙,0≤𝜙<∞. (105) In this way, the method developed in this paper may be used in order to give a new geometrical interpretation of rapidity widely used in relativistic physics (see, e.g., [12] and references therein).