Hyperbolic Geometry appeared in the first half of the 19th century as an attempt to understand Euclid's axiomatic basis of Geometry. It is also known as a type of non-Euclidean Geometry, being in many respects similar to Euclidean Geometry

Barbu, Catalin

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Scientia MagnaVol. 6 (2010), No. 1, 19-24Menelaus’s theorem for hyperbolicquadrilaterals in the Einstein relativisticvelocity model of hyperbolic geometryCa˘ta˘lin BarbuVasile Alecsandri College, Baca˘u, str. Iosif Cocea, nr. 12, sc. A, ap. 13, RomaniaE-mail: kafka mate@yahoo.comAbstract In this study, we present (i) a proof of the Menelaus theorem for quadrilaterals inhyperbolic geometry, (ii) and a proof for the transversal theorem for triangles, and (iii) theMenelaus’s theorem for n-gons.Keywords Hyperbolic geometry, hyperbolic triangle, hyperbolic quadrilateral, Menelaustheorem, transversal theorem, gyrovector.§1. IntroductionHyperbolic Geometry appeared in the first half of the 19th century as an attempt to un-derstand Euclid’s axiomatic basis of Geometry. It is also known as a type of non-EuclideanGeometry, being in many respects similar to Euclidean Geometry. Hyperbolic Geometry in-cludes similar concepts as distance and angle. Both these geometries have many results incommon but many are different. There are known many models for Hyperbolic Geometry, suchas: Poincare´ disc model, Poincare´ half-plane, Klein model, Einstein relativistic velocity model,etc. Menelaus of Alexandria was a Greek mathematician and astronomer, the first to recognizegeodesics on a curved surface as natural analogs of straight lines. Here, in this study, we presenta proof of Menelaus’s theorem for quadrilaterals, a proof for the transversal theorem, and aproof of Menelaus’s theorem for n-gons in the Einstein relativistic velocity model of hyperbolicgeometry. The well-known Menelaus theorem states that if l is a line not through any vertex ofa triangle ABC such that l meets BC in D, CA in E, and AB in F , then DBDC · ECEA · FAFB = 1 [1].F. Smarandache (1983) has generalized the Theorem of Menelaus for any polygon with n ≥ 4sides as follows: If a line l intersects the n-gon A1A2 . . . An sides A1A2, A2A3, . . . , and AnA1respectively in the points M1,M2, · · · , and Mn, then M1A1M1A2 · M2A2M2A3 · . . . · MnAnMnA1 = 1 [2].Let D denote the complex unit disc in complex z - plane, i.e.D = {z ∈ C : |z| < 1}.The most general Mo¨bius transformation of D isz → eiθ z0 + z1 + z0z= eiθ(z0 ⊕ z),20 Ca˘ta˘lin Barbu No. 1which induces the Mo¨bius addition ⊕ in D, allowing the Mo¨bius transformation of the disc tobe viewed as a Mo¨bius left gyrotranslationz → z0 ⊕ z = z0 + z1 + z0zfollowed by a rotation. Here θ ∈ R is a real number, z, z0 ∈ D, and z0 is the complex conjugateof z0. Let Aut(D,⊕) be the automorphism group of the grupoid (D,⊕). If we definegyr : D ×D → Aut(D,⊕), gyr[a, b] = a⊕ bb⊕ a =1 + ab1 + ab,then is true gyrocommutative lawa⊕ b = gyr[a, b](b⊕ a).A gyrovector space (G,⊕,⊗) is a gyrocommutative gyrogroup (G,⊕) that obeys the fol-lowing axioms:(1) gyr[u,v]a· gyr[u,v]b = a · b for all points a,b,u,v ∈G.(2) G admits a scalar multiplication, ⊗, possessing the following properties. For all realnumbers r, r1, r2 ∈ R and all points a ∈G:(G1) 1⊗ a = a.(G2) (r1 + r2)⊗ a = r1 ⊗ a⊕ r2 ⊗ a.(G3) (r1r2)⊗ a = r1 ⊗ (r2 ⊗ a).(G4) |r|⊗a‖r⊗a‖ =a‖a‖ .(G5) gyr[u,v](r ⊗ a) = r ⊗ gyr[u,v]a.(G6) gyr[r1 ⊗ v, r1 ⊗ v] =1 .(3) Real vector space structure (‖G‖ ,⊕,⊗) for the set ‖G‖ of onedimensional ”vectors”‖G‖ = {±‖a‖ : a ∈ G} ⊂ Rwith vector addition ⊕ and scalar multiplication ⊗, such that for all r ∈ R and a,b ∈ G,(G7) ‖r ⊗ a‖ = |r| ⊗ ‖a‖.(G8) ‖a⊕ b‖ ≤ ‖a‖ ⊕ ‖b‖.Definition 1. Let ABC be a gyrotriangle with sides a, b, c in an Einstein gyrovectorspace (Vs,⊕,⊗), and let ha, hb, hc be three altitudes of ABC drawn from vertices A,B,Cperpendicular to their opposite sides a, b, c or their extension, respectively. The numberSABC = γaaγhaha = γbbγhbhb = γccγhchcis called the gyrotriangle constant of gyrotriangle ABC (here γv = 1√1− ‖v‖2s2is the gammafactor). (see [3, pp558])Theorem 1. (The Gyrotriangle Constant Principle) Let A1BC and A2BC be twogyrotriangles in a Einstein gyrovector plane (R2s,⊕,⊗), A1 6= A2 such that the two gyrosegmentsA1A2 and BC, or their extensions, intersect at a point P ∈ R2s, as shown in Figs 1-2. Then,γ|A1P | |A1P |γ|A2P | |A2P |=SA1BCSA2BC.Vol. 6Menelaus’s theorem for hyperbolic quadrilaterals in the Einsteinrelativistic velocity model of hyperbolic geometry 21([3, pp 563])Theorem 2. (The Hyperbolic Theorem of Menelaus in Einstein GyrovectorSpace) Let a1,a2, and a3 be three non-gyrocollinear points in an Einstein gyrovector space(Vs,⊕,⊗). If a gyroline meets the sides of gyrotriangle a1a2a3 at points a12,a13,a23, as inFigure 3, thenγªa1⊕a12 ‖ªa1 ⊕ a12‖γªa2⊕a12 ‖ªa2 ⊕ a12‖γªa2⊕a23 ‖ªa2 ⊕ a23‖γªa3⊕a23 ‖ªa3 ⊕ a23‖γªa3⊕a13 ‖ªa3 ⊕ a13‖γªa1⊕a13 ‖ªa1 ⊕ a13‖= 1.(see [3, pp 463])For further details we refer to the recent book of A.Ungar [3].§2. Menelaus’s theorem for hyperbolic quadrilateralsIn this section, we prove Menelaus’s theorem for hyperbolic quadrilateral.Theorem 3. If l is a gyroline not through any vertex of a gyroquadrilateral ABCD suchthat l meets AB in X, BC in Y , CD in Z, and DA in W , thenγ|AX||AX|γ|BX||BX|· γ|BY ||BY |γ|CY ||CY |· γ|CZ||CZ|γ|DZ||DZ|· γ|DW ||DW |γ|AW ||AW |= 1. (1)Proof of Theorem 3. Let T be the intersection point of the gyrolineDB and the gyrolineXY Z (See Figure 4). If we use a Theorem 2 in the triangles ABD and BCD respectively, then22 Ca˘ta˘lin Barbu No. 1γ|AX||AX|γ|BX||BX|· γ|BT ||BT |γ|DT ||DT |· γ|DW ||DW |γ|AW ||AW |= 1, (2)andγ|DT ||DT |γ|BT ||BT |· γ|CZ||CZ|γ|DZ||DZ|· γ|BY ||BY |γ|CY ||CY |= 1. (3)Multiplying relations (2) and (3) member with member, we obtain the conclusion.§3. The hyperbolic transversal theorem for trianglesIn this section, we prove the hyperbolic transversal theorem for triangles.Theorem 4. Let D be on gyroside BC, and l is a gyroline not through any vertex of aVol. 6Menelaus’s theorem for hyperbolic quadrilaterals in the Einsteinrelativistic velocity model of hyperbolic geometry 23gyrotriangle ABC such that l meets AB in M, AC in N , and AD in P , thenγ|AM||AM |γ|AB||AB|· γ|AC||AC|γ|AN||AN |· γ|PN||PN |γ|PM||PM |· γ|DB||DB|γ|DC||DC|= 1.Proof of Theorem 3. If we use a theorem 3 for gyroquadrilateral BCNM and gyro-collinear points D,A, P , and A (See Figure 5) then the conclusion follows.§4. Menelaus’s Theorem for n - gonsIn this section, we prove Menelaus’s theorem for hyperbolic n− gons.Theorem 5. If l is a gyroline not through any vertex of a n − gyrogon A1A2...An suchthat l meets A1A2 in M1, A2A3 in M2, ..., and AnA1 in Mn, thenγ|M1A1||M1A1|γ|M1A2||M1A2|· γ|M2A2||M2A2|γ|M2A3||M2A3|· ... · γ|MnAn||MnAn|γ|MnA1||MnA1|= 1. (4)Proof of Theorem 5. We use mathematical induction. For n = 3 the theorem is true(see Theorem 2). Let’s suppose by induction upon k ≥ 3 that the theorem is true for anyk− gyrogon with 3 ≤ k ≤ n−1, and we need to prove it is also true for k = n. Suppose a line lintersect the gyroline A2An into the point M . We consider the n− gyrogon A1A2...An and wesplit in a 3− gyrogon A1A2An and (n− 1)− gyrogon AnA2A3...An−1 and we can respectivelyapply the theorem 2 according to our previously hypothesis of induction in each of them, andwe respectively get:γ|M1A1||M1A1|γ|M1A2||M1A2|· γ|MA2||MA2|γ|MAn||MAn|· γ|MnAn||MnAn|γ|MnA1||MnA1|= 1,andγ|MAn||MAn|γ|MA2||MA2|· γ|M2A2||M2A2|γ|M2A3||M2A3|· ... ·γ|Mn−2An−2||Mn−2An−2|γ|Mn−2An−1||Mn−2An−1|·γ|Mn−1An−1||Mn−1An−1|γ|Mn−1An||Mn−1An|= 1.24 Ca˘ta˘lin Barbu No. 1whence, by multiplying the last two equalities, we getγ|M1A1||M1A1|γ|M1A2||M1A2|· γ|M2A2||M2A2|γ|M2A3||M2A3|· ... · γ|MnAn||MnAn|γ|MnA1||MnA1|= 1.AcknowledgementThe author would like to thank to Florentin Smarandache for helpful suggestions.References[1] R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry,Washington, DC: Math. Assoc. Amer., 147-154, 1995.[2] F. Smarandache, Ge´ne´ralisation du The´orcˇme de Me´ne´laus, Rabat, Morocco, Seminarfor the selection and preparation of the Moroccan students for the International Olympiad ofMathematics in Paris-France, 1983.[3] A. A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory ofRelativity, Hackensack, NJ: World Scientific Publishing Co.Pte. Ltd., 2008.

Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity,

Episodes

Ge´ne´ralisation du The´orcˇme de Me´ne´laus, Rabat, Morocco, Seminar for the selection and preparation of the Moroccan students for the International Olympiad of Mathematics in Paris-France,

Menelaus's theorem for hyperbolic quadrilaterals in the Einstein relativistic velocity model of hyperbolic geometry

https://zenodo.org/record/9307/files/MenelausTheorem.pdf

Menelaus's theorem for hyperbolic quadrilaterals in the Einstein relativistic velocity model of hyperbolic geometry

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