Hyperbolic cosines and sines theorems for the triangle formed by intersection of three semicircles on Euclidean plane

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1 Hyperbolic cosines and sines theorems for the triangle formed by intersection of three semicircles on Euclidean plane arxiv:04.535v [math.gm] 3 Apr 0 Robert M. Yamaleev Facultad de Estudios Superiores Universidad Nacional Autonoma de Mexico Cuautitlán Izcalli Campo C.P México. Joint Institute for Nuclear Research LIT Dubna Russia. iamaleev@servidor.unam.mx April 8 0 Abstract The purpose the present paper is to construct the hyperbolic trigonometry on Euclidean plane without refereing to hyperbolic plane. In this paper we show that the concept of hyperbolic angle and its functions forming the hyperbolic trigonometry give arise on Euclidean plane in a natural way. The method is based on a key- formula establishing a relationship between exponential function and the ratio of two segments. This formula opens a straightforward pathway to hyperbolic trigonometry on the Euclidean plane. The hyperbolic law of cosines I and II and the hyperbolic law of sines are derived by using of the key-formula and the methods of Euclidean Geometry only. It is shown that these laws are consequences of the interrelations between distances and radii of the intersecting semi-circles. Introduction Traditionally interrelations between angles and sides of a triangle are described by the trigonometry via periodic sine-cosine functions. The concept of the angle in Euclidean plane is intimately related with the figure of a circle and with motion of a point along the circumference. The hyperbolic trigonometry also intimately is related with the circle reflecting the hyperbolic properties of the circles. The properties of a triangle formed by intersection of three semicircles plays a principal role in the upper half-plane model of the hyperbolic plane( Lobachevskii plane). There are several models for the hyperbolic plane [] however all these models use the same idea as upper half-plane model H [3] []. The geometry of the hyperbolic plane within the framework of H model is studied by considering quantities invariant under an action of the general Möbius group which consists of compositions of Möbius transformations and reflections [4]. The model of the hyperbolic plane is the upper half-plane model. The underlining space of this model is the upper half-plane H in the complex plane C defined to be H {z C : Im(z) > 0}.

2 It is used the usual notion of point that H inherits from C. It is also used the usual notion of angle that H inherits from C that is the angle between two curves in H is defined to be the angle between the curves when they are considered to be curves in C which in turn is defined to be the angle between their tangent lines. Usually it is thought that the hyperbolic laws of cosines and sines in the upper half-plane model are consequences of a special structure of the complex plane. The purpose the present paper is to construct the hyperbolic trigonometry on Euclidean plane without refereing to hyperbolic plane. The principal goal of the paper is to show that the concept of hyperbolic angle and its functions forming the hyperbolic trigonometry give arise on Euclidean plane in a natural way. Our method is based on one simple but useful formula of the hyperbolic calculus which we denominated as a Key-formula. The Key-formula establishes relationship between exponential function and the ratio of two quantities. This formula opens the straightforward pathway to hyperbolic trigonometry on Euclidean plane. It is well-known the hyperbolic law of sines and the hyperbolic law of cosines II are derived from the hyperbolic law of cosines I by algebraic manipulation [3] []. In this paper we prove all three laws separately on making use of interrelations between distances between centers and radii of circles on Euclidean plane only. In Section the Key-formula of hyperbolic calculus is substantiated. On making use of the Key-formula elements of the right-angled triangle are expressed via hyperbolic trigonometry. In Section 3 two main relationships between elements of the triangle formed by intersections of semicircles is established. In Section 4 the theorem of cosines for the triangle formed by three intersecting semicircles denominated as hyperbolic law of cosines I is proved. In Section 5 the hyperbolic law of sines and the hyperbolic law of cosines II are derived on making use of the main relationships between thee circles. Hyperbolic trigonometry in Euclidean Geometry. Key-formula of hyperbolic calculus. The Key-formula of hyperbolic calculus has been established in Ref.[5]. Let a a(s) b b(s) are real functions of the parameter s. If the difference (a b) does not depend of the parameter s then the following formula holds true exp((a b)s) a b. (.) The formula (.) constitutes a substance what we shall refer as Key-formula. The main advantage of the Key-formula is the following: the argument of the exponential function is proportional to the difference between the nominator and the denominator if this difference is a constant. If (a b) does not depend of s then functions a and b can be presented as follows a a 0 +Y(s) b b 0 +Y(s) where a 0 b 0 are constants. For the ratio of these constants find φ 0 satisfying the equation exp((a 0 b 0 )φ 0 ) a 0 b 0. Since a shift of the pair {a 0 b 0 } by Y does not change (a 0 b 0 ) one may write a 0 +Y b 0 +Y a b exp((a 0 b 0 )(φ 0 +δ)) exp((a b)φ). (.)

3 . Elements of right-angled triangle as functions of a hyperbolic argument. Let ABC be a right-angled triangle with right angle at C. Denote the sides by ab the hypotenuse by c the angles opposite by A B C correspondingly. Circular functions of the angles are defined in the usual way. For instance sinb b c tanb b a. (.3) These two ratios are functions of the same angle B. Our purpose is to introduce an hyperbolic angle in a such way that these ratios will be expressed as functions of unique hyperbolic angle. This achieved as follows. Define the hyperbolic angle ξ by the following relation c+a c a exp(ξ). (.4) From this equation it follows a c tanh(ξ) a sinh(ξ). (.5) b This leads to the following interrelations between circular and hyperbolic trigonometry cosb a c tanh(ξ) tana b a sinh(ξ). (.6) Now let us explore the same problem by using a geometrical motion. We shall change sides b and c remaining unchanged the side a and the right angle C. Since the length a is a constant of this motion in agreement with Key-formula (.) we write c+a c a exp(aφ). (.7) Notice now the argument of exponential function in (.6) is proportional to a: ξ aφ where φ is a parameter of the evolution. Formulae (.6) are re-written as cosb a c tanh(aφ) tanb b a sinh(aφ). (.8) It is interesting to observe that within these representation we are able to find a limit for a 0. In fact at this limit we get c(a 0) b(a 0) φ. The relations between circular and hyperbolic trigonometry (.8) also may be put into other form for instance cotb sinh(aφ) (.9) or tan B exp( aφ). (.0) Install the triangle ABC in such a way that the side b lies on horizontal axis X the side a is perpendicular to this line at the point C (Fig.). The line c cuts X in A. The point of intersection A may continue moving and the distance AC tends to infinity. Let us recall the problem of parallel lines in Geometry. 3

4 The ray AB then tends to a definite limiting position BL and BL is said to be parallel to X. As the point A moves along X away from C there are two possibilities to consider: () In Euclidean Geometry the angle between two lines BL and BC is equal to right angle. () The hypothesis of Hyperbolic Geometry is that this angle less than the right angle. The most fundamental formula of the Hyperbolic Geometry is the formula connecting the parallel angle Π(d) and the length d. In order to establish those relationship the concept of horocycles some circles with center and axis at infinity were introduced [6]. 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 parallel angle with the distance d is given by exp( d Π(p) ) tan κ. (.) Now introduce in (.0) the value inverse to φ K and write (.0) in the form φ exp( a K ) tan B. (.) Let the point to tend to infinity. The following two cases can be considered. () φ will tend to zero K and B π. This is true in Euclidean Geometry. () Suppose that φ K and the angle B go to some limited values lim K κ limb Π(a). (.3) AC In this way from (.0) we come to main formula of hyperbolic geometry. 3. Rotational motion of a line tangent to the semicircle. Let us explore a motion of the line L around the semicircle C remaining tangent to the semicircle [5]. Draw semicircle C (Fig.) end-points and the center of which lie on horizontal axis X-axis. Denote by B center of the semicircle and by K K end-points of the semicircle on X-axis. Draw the line L tangent to the semicircle at the point C. This line intersects with X-axis at the point A. Through end-points of the semicircle K K erect the lines parallel to vertical axis Y-axis. The intersections of these lines with line L denote by P and P correspondingly. Draw line parallel to Y-axis from the center B which acroses C at the point N at the top of the semicircle C. This line intersects with the line L at the point M. From point M draw horizontal line which acroses the vertical line K P at the point M. Denote by r radius of the circle so that r BN BC. Denote by B the angle between AB and BC. The angle C is rectangle so that (AB) (AC) r. (.4) Consider rotational motion of the line L around semicircle C remaining tangent to C where the point C runs between points K and N. During of the motion the length of segments AC and AB will change but the triangle remains to be right-angled. This is exactly the case considered above difference is that now ABC installed in another position with respect to horizontal X-axis. We have seen this evolution process is described by equations AB rcoth(ξ) AC 4 r sinh(ξ) (.5)

5 where ξ rφ. From similarity of triangles ABC and AK P we find Consequently Then obviously Furthermore P K AP r AC sinh(ξ). (.6) P K AK sinh(ξ) (AB r)sinh(ξ) rexp( ξ). (.7) P K rexp(ξ). (.8) BM P K +P K rcosh(ξ) P M rsinh(ξ) (.9) Thus all segments of the lines in Fig. can be expressed via hyperbolic angle ξ and the radius of the semicircle r. On the basis of obtained formulae the following relationships between circular and hyperbolic trigonometric functions are established sin(b) cot(b) sinh(ξ) cos(b) tanh(ξ). (.0) cosh(ξ) 3 Relationships between elements of three intersecting semicircles 3. Hyperbolic cosine- sine functions of arcs of the semicircle. Up till now we were able to define hyperbolic trigonometric functions of the arcs originated from the top N ( for variable ξ). Now let us calculate trigonometric functions of the arcs with arbitrary end-points on the semicircle. Consider arc A A defined in the first quadrant of the semicircle with end-points at the points A and A where NA < NA. Since we are able to calculate arcs with the origin installed on the top of the semicircle ( for variable ξ) we shall present this segment on the circle as difference of two segments both originated from the top of the semicircle. For example A A NA NA. Then cosh A A cosh NA cosh NA sinh NA sinh NA sinh A A sinh NA cosh NA cosh NA sinh NA. (3.) Denote by a a the angles formed by radiuses O a A and O a A with X-axiscorrespondingly. Then the functions coshna coshna sinhna sinhna are expressed via circular trigonometric functions as follows cosh NA sina cosh NA sina sinh NA cota sinh NA cota. (3.) On making use of equations (3.) in (3.) we get cosha A cosa cosa sinha A cosa cosa. (3.3) sina sina sin sina 5

6 3. The main relationships between elements of the triangle formed by intersections of semicircles. In Fig.3 three intersecting semicircles with centers installed on horizontal axis at the points O a O b O c are presented. Intersections of the circumferences form triangle ABC bounded by segments of the circumferences a BC c ÂBb ÂC. Connect vertex of the triangle with centers of the circle by corresponding radiuses. Denote by a k b k c k k the angles bounded by the radiuses and X-axis where a > a > 0b > b > 0c > c > 0. By making use of (3.3) define hyperbolic cosine-sine functions corresponding to bounding segments cosha cosa cosa sina sina sinha cosa cosa sina sina coshb cosb cosb sinb sinb sinhb cosb cosb sinb sinb (3.4a) (3.4b) coshc cosc cosc sinhc cosc cosc. (3.4c) sinc sinc sinc sinc It is used the usual notion of the angle that is the angle between two curves is defined as an angle between their tangent lines. Let the angles αβγ be angles at the vertex ABC correspondingly. For these angles we can define its proper cosine and sine functions. The angles of the triangle ABC αβγ are closely related with angles a a b b c c. From draught in Fig.3 we find the following relationships between them β a +c δ π a b α b c. (3.5) Then cosα cosb cosc +sinb sinc cosβ cosa cosc sina sinc cosδ cosb cosa +sinb sina sinα sinb cosc cosb sinc sinβ sina cosc +cosa sinc sinδ sinb cosa +cosb sina. Denote distances between centers by (3.6a) (3.6b) (3.6c) (3.7a) (3.7b) (3.7c) O cb O c O b O ba O b O a O ac O a O c. The theorem of sines employed for triangles O c AO b O b CO a O c BO a gives six relations of type sinα O cb sinc r b sinb r c (3.8a) sinγ sina sinb (3.8b) O ba r b r a sinβ sinc sina. (3.8c) O ac r a r c From these relations it follows the first set of main relationships: Relation I. r a sina r b sinb r c sinc r a sina r c sinc r b sinb (3.9) 6

7 From the draught in Fig.3 it is seen that O ac O ba +O cb (3.0) where Hence O ac r c cosc +r a cosa O ba r a cosa +r b cosb. (3.) O cb r c cosc +r a cosa r a cosa r b cosb. (3.) From vertices of ABC erect lines perpendicular to horizontal line which intersect X-axis at points h A h B h C correspondingly. From the draught in Fig.3 we find that O cb O c h A h A O b r c cosc r b cosb. (3.3) By equating (3.) with (3.3) we arrive to another main relationship between radii and angles: Relation II. r c cosc r b cosb r c cosc +r a cosa r a cosa r b cosb. (3.4) We shall effect a simplification by using the following designations. r a cosa w 0 r a sina w r a cosa v 0 r a sina v r b cosb w 0 r b sinb w r b cosb v 0 r b sinb v r c cosc w 03 r c sinc w 3 r c cosc v 03 r c sinc v 3. In these designations formulae (3.6abc) and (3.7abc) are written as follows r b r c sinα w w 03 w 0 w 3 r b r c cosα w 0 w 03 +w w 3 r a r c sinβ v v 03 +v 0 v 3 r a r c cosβ v 0 v 03 v v 3 r a r b sinδ v w 0 +v 0 w r a r b cosδ v w v 0 w 0 (3.5a) (3.5b) (3.5c) Formulae (3.4abc) for hyperbolic sines and cosines are re-written as follows cosha r a w 0 v 0 w v sinha r a w 0 v 0 w v coshb r b w 0v 0 w v sinhb r b w 0 v 0 w v coshc r c w 03v 03 w 3 v 3 sinhc r c w 03 v 03 w 3 v 3. (3.6) The equations of main Relation I now take the form Equations (3.8)-(3.) are re-written as follows x : w w 3 y : v v 3 z : w v. (3.7) O cb w 03 w 0 O ac v 03 +v 0 O ba w 0 +v 0. 7

8 Correspondingly the main Relation II takes the form w 03 w 0 v 03 +v 0 w 0 v 0. (3.8) This expresses the fact that O ca is a sum of O cb and O ba. Notice equation (3.8) can be re-written also in another equivalent form namely w 03 v 03 w 0 v 0 (w 0 v 0 ). (3.9) Denote the segments -projections of sides of ABC on X-axis by P(AC) h A h C P(AB) h A h B P(BC) h B h C. From the draught in Fig.3 it is seen that P(AC) P(AB)+P(BC) (3.0) where P(BC) w 0 v 0 P(AC) w 0 v 0 P(AB) w 03 v 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 triangle ABC formed by intersection of semicircles with centers installed on X-axis (Fig.3) which is given by the set of three equations coshc coshacoshb sinhasinhbcosδ coshb coshacoshc sinhcsinhacosβ cosha coshccoshb sinhcsinhbcosα. Theorem of cosines I. The following equation for elements of the triangle ABC formed by intersection of three circles holds true coshc coshacoshb sinhasinhbcosδ. (4.) Proof Square both sides of the main Relation II to obtain (w 03 v 03 ) (w 0 v 0 ) +(w 0 v 0 ) (w 0 v 0 )(w 0 v 0 ) (4.) and evaluate this equality by taking into account formulae (3.5)-(3.6). First of all evaluate the left-hand side of this as follows and notice that v 03 +w 03 v 03w 03 v 03 +w 03 r c +(r c v 03w 03 ) (4.3) Transform (4.3) into the following form r c w 03 +w 3 +v 03 +v 3. (4.4) v03 +w 03 r c v 03 +w 03 (w 03 +w 3 +v 03 +v 3 ) (w 3 +v 3 ). (4.5) 8

9 Equation (4.) is written as (w 3 +v 3 ) }{{} +(r c v 03w 03 ) (w 0 v 0 ) +(w 0 v 0 ) (w 0 v 0 )(w 0 v 0 ). (4.6) The underlined term pass from the left-hand side to the right-hand side of the equation. Then in the left-hand side we remain with the expression We arrive to the following equation (r c v 03 w 03 ) v 3 w 3 coshc. (4.7) v 3 w 3 coshc v 3 w 3 ( w 3 +v 3 +(v 0 w 0 ) +(v 0 w 0 ) (v 0 w 0 )(v 0 w 0 ) ). (4.8) From the second main Relation I we use w 3 w v 3 v. This makes true the following equation v w. (4.9) w 3 v 3 w v w v The factor of the right-hand side of (4.8) replace by the right-hand side of (4.9). In this way we come to the following equation coshc {v w (w3 w v w v +v 3 +(v 0 w 0 ) +(v 0 w 0 ) )} v w (v 0 w 0 )(v 0 w 0 ). w v w v } {{} (4.0) Evaluate now underlined term in the right-hand side of equation (4.0) which we firstly transform as follows (v 0 w 0 )(v 0 w 0 ) v w v w r b (v 0 w 0 ) r a (v 0 w 0 ). (4.) w v w v r a r b w v w v Then use the second of equations of (3.5c) written as v w r a r b cosδ + v 0w 0 r a r b. (4.) By making use of equation (4.) we are able to evaluate equation (4.) as follows v w r a r b r b (v 0 w 0 ) w v r a (v 0 w 0 ) w v ( cosδ + v 0w 0 r a r b ) r b(v 0 w 0 ) w v r a (v 0 w 0 ) w v cosδ r b(v 0 w 0 ) w v r a (v 0 w 0 ) w v v 0 w 0 (v 0 w 0 ) w v (v 0 w 0 ) w v } cosδ sinhasinhb {{} v (v 0 w 0 )(v 0 w 0 ) 0w 0. (4.3) w v w v Replace underlined term of (4.0) by (4.3) and pass the underlined expression of (4.3) to the left-hand side of obtained equation. As a result we come to the following equation coshc cosδ sinhbsinha 9

10 w v w v { v w ( w +v +(w 0 v 0 ) +(w 0 v 0 ) ) v 0 w 0 (w 0 v 0 )(w 0 v 0 ) }. (5.4) On making use of elementary algebra one may show that (see the section Appendix) v w ( w +v +(w 0 v 0 ) +(w 0 v 0 ) ) v 0 w 0 (w 0 v 0 )(w 0 v 0 ) (r a v 0w 0 )(r b v 0w 0 ). (4.5) Now substitute (4.5) into (4.4) and take into account (3.6). This gives (ra w v v w v 0w 0 )(rb v 0w 0 ) cosha coshb (4.6) by using of which we arrive from (4.4) to the following equation coshc coshacoshb cosδsinhasinhb. (4.7) End of Proof. The other two equations of the law obviously are proved analogously. 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 sinα sinhb sinβ sinhc sinδ (5.) and the Hyperbolic law (theorem) of cosines II given by the formulae cosδ cosαcosβ sinαsinβcoshc (5.) cosβ cosαcosδ coshbsinαsinδ (5.3) cosα cosβcosδ coshasinδsinβ. (5.4) 5. Hyperbolic theorem of sines and its geometrical interpretation on Euclidean plane. Lemma 5. 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 a cosa r a cosa P(AC) r b cosb r b cosb P(AB) r c cosc r c cosc. (5.5) Distances between centers of the circles have been defined as (see (3.) (3.)) and O ca r c cosc +r a cosa O ba r a cosa +r b cosb O cb r c cosc r b cosb (5.6) P(AC) P(BC)+P(AB) O ca O bc +O ab. (5.7) Write the first main Relation I given by the set of equations r a sina r b sinb r c sinc r a sina r c sinc r b sinb (5.8) 0

11 in a squared form namely r a r a cos a r b r b cos b r c r c cos c r a r a cos a r c r c cos c r b r b cos b. (5.9) Then for the squared distances O ikik abc we have O ca r c cos c +r a cos a +r c cosc r a cosa O ba r acos a +r b cos b +r a cosa r b cosb O cb r c cos c +r b cos b r c cosc r b cosb. (5.0) Combine equations (5.9) with (5.0) this leads to the following system of equations (a) O ca r c r a +r acosa O ac (b) O ca r a r c +r ccosc O ac (a) O ba r a r b +r bcosb O ba (b) O ba r b r a +r acosa O ba (a) O cb r b r c +r ccosc O cb (b) O cb r c r b r bcosb O cb. (5.a) (5.b) (5.c) From these equations the cosines of the angles a a b b c c are expressed: O ca r c +r a r a O ac cosa O ba r a +r b r b O ba cosb O ca r a +r c r c O ac cosc O ba r b +r a r a O ba cosa Ocb r b +r c Ocb cosc +r c r b cosb. (5.) r c O cb r b O cb Having these formulae we may present the projection P ca as follows P ac r b cosb r b cosb O cb +r c r b O cb O ba r a +r b O ba ( O cb +r c r b )O ba (O ba r a +r b )O cb O cb O ba O cb O ba +r c O ba r b O ba ( O ba r a +r b ) O cb O cb O ba O cbo ba +r co ba r b O ba O bao cb +r ao cb r b O cb O cb O ba The first ratio is presented as follows Π ca O ca O cbo ba ( O cb + O ba )+r c O ba r b (O ba O cb )+r a O cb O cb O ba O ca O cb O ba ( O ca )+r c O ba r b (O ca)+r a O cb O cb O ba O ca. (5.3) Now in the same way let us calculate the next ratio Π bc O bc. Formula for the projection evaluated as follows P bc r a cosa r a cosa

12 O ba r b +r a O ba O ca r c +r a O ac ( O ba r b +r a ) O ac ( O ca r c +r a )O ba O ac O ba ( O bao ac r bo ac +r a O ac ) ( O cao ba r co ba +r a O ba ) O ac O ba ( O ba O ac (O ba O ca ) r b O ac +r a O ac )+r c O ba r a O ba ) O ac O ba. Take into account O cb O ba O ac hence Now calculate the ration P bc O ba O ac (O cb r bo ac +r a (O cb )+r co ba ) O ac O ba ( O ba O ac (O cb ) r b O ac +r a (O cb)+r c O ba ) O ac O ba. P bc ( O bao ac (O cb ) rbo ac +ra (O cb )+rco ba ). (5.4) O bc O ac O ba O bc This expression coincides with (5.3) consequently By taking into account (5.7) we arrive to the desired relations P bc O bc P ca O ca. (5.5) P(BC) O bc P(AC) O ca P(AB) O ab. (5.6) End of proof. Now come back to designations introduced in Section 3. In these designations equations (5.6) are written as follows w 0 v 0 w 03 w 0 w 0 v 0 v 03 +v 0 w 03 v 03 w 0 +v 0. (5.7) Theorem 5. The sides and the angles of triangle ABC satisfy the equations (5.). Proof By using the designations introduced in Sec. the system of equations (6.) can be written as follows sinha sinα r a(w 0 v 0 ) yz sinhb sinβ w 0 v 0 r b xz sinhc sinδ w 03 v 03 r c yx : : : x ( w 03 w 0 ) r b r c w 0 v 0 w 03 w 0 y ( v 03 +v 0 ) r a r c w 0 v 0 v 03 +v 0 z ( w 0 +v 0 ) r b r a w 03 v 03 w 0 +v 0 xyz r a r b r c xyz r a r b r c xyz r a r b r c. (5.8)

13 It is seen these equations contain a common factor which is symmetric with respect to abc and xyz. Multiply all equations (5.7) by this factor. We arrive to equations (5.). End of proof Theorem 5.3. The sides and the angles of triangle ABC satisfy the following equation. cosδ sinαsinβcoshc cosαcosβ. (5.9) Proof. Evaluate the first term of the right-hand side of (5.9). sinαsinβcoshc r a r b O ca O cb ( cosc cosc ) r a r b O ca O cb ( O cb r b +r c r c O cb O ca O cb O O ca O cb r b +rc cb r a r b r a r b r c O cb O ca O cb r a r b 4r a r b rc O ca r a +r c r c O ac ) O ca r a +r c r c O ac ) ( O cb r b +r c ) ( O ca r a +r c ) ( 4O 4r a r b rc ca O cb rc ( Ocb rb +rc ) ( Oca ra +rc ) ) ( 4O 4r a r b rc ca O cb rc O cb O ca ( O cb +O ca )r c +O cb r a +O ca r b ). (5.0) Now calculate the product cosαcosβ by using the following formulae We get cosα (r c r b r +r b O cb ) cosβ (rc c r a r +r a O ca ). c cosαcosβ r b r c (r c +r b O cb) r a r c (r c +r a O ca) ( O 4r b r a rc cb O ca ( O cb +O ca )r c O cb r a O ca r b ). (5.) By using equations (5.0) and (5.) calculate the difference 4r a r b r c Thus we got the equation (5.9). End of proof. sinαsinβcoshc cosαcosβ ( 4O ca O cb r c O cbo ca ( O cb+o ca)r c +O cbr a +O car b) ( +O 4r b r a rc cb O ca ( O cb +O ca )r c O cb r a O ca r b ) (ra r b r +r b O ab ) a cosδ. (5.) 3

14 The other two equations (5.3)and (5.4) are proved analogously. Concluding remarks. We have seen that the hyperbolic trigonometry like circular angle gives arise in a natural way on the Euclidean plane. The hyperbolic description of the elements of the Euclidean plane has to be considered as a complementary tool of the Euclidean Geometry. This description provides with new insights into hidden nature of the Euclidean Geometry. The proofs of theorems hyperbolic law of cosines I hyperbolic law of sines and hyperbolic law od cosines II were based purely on elements of the Euclidean geometry. These laws express interrelations between distances of the circles radii and angles between radiuses and X-axis. The method developed in this paper opens new pathway from Euclidean to hyperbolic geometry and can be used as an introduction into complex field of hyperbolic geometry. 6 Appendix The task of this section is to reduce the expression v w ( w +v +(w 0 v 0 ) +(w 0 v 0 ) ) v 0 w 0 (w 0 v 0 )(w 0 v 0 ) (A.) onto the expression (r a v 0w 0 )( r b v 0w 0 ). (A.) By taking into account the equation v w and opening the brackets transform (A.) into the following form ww + v v + w w 0 +w v 0 w 0v 0 w + w0 v + v 0v v 0 w 0 v }{{} + v 0 w 0 w 0 v 0 v 0 w 0 w 0 +v 0 w 0 v 0 w 0v 0 }{{}. (A.3) Joint together terms marked by same under- and over- lines and take into account that w w +w w 0 w r b vv + v 0v v ra w 0 v 0 w v 0w 0 w 0 v 0w 0 r a v 0 w 0 v }{{} v 0w 0 v 0 }{{} v 0w 0 r b. The last term in (A.3) represent as follows (v0 w 0 v 0 w 0 +v 0 w 0 ). In this way we transform expression (A.3) into the following form v 0 w 0 w 0 v 0 4

15 v 0 w 0 r a v 0w 0 r b + w rb +v ra+ {}}{ w v0 }{{} + w0 v+ {}}{ v0 w 0 +v 0 w 0 }{{}. (A.4) Join term with same under- and over- lines taking into account that {}}{ w 0 v + {}}{ v 0 w 0 w 0 r b w v 0 }{{} +v 0 w 0 }{{} v 0 r a. Then we fulfil to the following set of simple transformations v 0 w 0 w 0 v 0 v 0 w 0 r a v 0w 0 r b +w r b +v r a +v 0 r a +w 0 r b v 0 w 0 w 0 v 0 + End of proof. v 0 w 0 r a v 0w 0 r b r a r b (r a v 0 w 0 )( r b v 0 w 0 ) References [] S.Stahl The Poincaré Half-Plane Jones and Bartlett Boston 993. [] Hyperbolic Geometry. Springer-Verlag London-Berlin-Heidelberg.Birkhauser ISBN [3] S.Katok Fuchsian Groups. Chicago Lectures in Mathematics University of Chicago Press IL 99. ISBN [4] A.A.Ungar Hyperbolic trigonometry and its applications in the Poincaré ball model of hyperbolic geometry Comput.Math.Appl. 4(/) (00) [5] R.M.Yamaleev Geometrical and physical interpretation of evolution governed by general complex algebra J.Math.Anal.Appl. (008)doi:0.06/ j.jmaa R.M. Yamaleev Geometrical and physical interpretation of the evolution governed by mass-shell equation. J.of Phys.:Conference Series 66 (007) 006. [6] N.J.Lobatschewsky Etudes géométriques sur la théorie de parallèles suivis d un extrait de la correspondance de Gauss et de Schumacher. traduit par G.J.Hoüel. Mémoires de la Société des sciences physiques et naturelles de Bordeaux. 4 Bordeaux