# Van Aubel's theorem in the einstein relativistic velocity model of hyperbolic geometry.

1 Introduction

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. Hyperbolic Geometry includes similar concepts as distance and angle. Both these geometries have many results in common but many are different. There are known many models for Hyperbolic Geometry, such as: Poincare disc model, Poincare half-plane, Klein model, Einstein relativistic velocity model, etc. Here, in this study, we give hyperbolic version of Van Aubel theorem. The well-known Van Aubel theorem states that if ABC is a triangle and AD, BE, CF are concurrent cevians at P, then AP/PD = AE/EC + AF/FB (see [1, p. 271]).

Let D denote the complex unit disc in complex z - plane, i.e.

D = {z [member of] C : [absolute value of (z)] < 1}.

The most general Mobius transformation of D is

z [right arrow] [e.sup.i[theta]] [z.sub.0] + z/1 + [bar.[z.sub.0]]z = [e.sup.i[theta]]([z.sub.0] [direct sum] z),

which induces the Mobius addition [direct sum] in D, allowing the Mobius transformation of the disc to be viewed as a Mobius left gyrotranslation

z [right arrow] [z.sub.0] [direct sum] z = [z.sub.0] + z/1 + [bar.[z.sub.0]]z

followed by a rotation. Here [theta] [member of] R is a real number, z, [z.sub.0] [member of] D, and [bar.[z.sub.0]] is the complex conjugate of [z.sub.0]. Let Aut(D, [direct sum]) be the automorphism group of the grupoid (D, [direct sum]). If we define

gyr : D x D [right arrow] Aut(D, [direct sum]), gyr[a, b] = a [direct sum] b/b [direct sum] a = 1 + a[bar.b]/1 + [bar.a]b,

then is true gyrocommutative law

a [direct sum] b = gyr[a, b](b [direct sum] a).

A gyrovector space (G, [direct sum], [direct sum]) is a gyrocommutative gyrogroup (G, [direct sum]) that obeys the following axioms:

1. gyr[u, v]a x gyr[u, v]b = a x b for all points a, b, u, v [member of] G.

2. G admits a scalar multiplication, [direct sum], possessing the following properties. For all real numbers r, [r.sub.1], [r.sub.2] [member of] R and all points a [member of] G:

(G1) 1 [cross product] a = a

(G2) ([r.sub.1] + [r.sub.2]) [cross product] a = [r.sub.1] [cross product] a [member of] [r.sub.2] [cross product] a

(G3) ([r.sub.1] [r.sub.2]) [cross product] a = [r.sub.1] [cross product] ([r.sub.2] [cross product] a)

(G4) [absolute value of (r)] [cross product] a/[parallel]r [cross product] a[parallel] = a/[parallel]a[parallel]

(G5) gyr[u, v](r [cross product] a) = r [cross product] gyr[u, v]a

(G6) gyr[[r.sub.1] [cross product] v, [r.sub.1] [cross product] v] =1

3. Real vector space structure ([parallel]G[parallel], [direct sum], [cross product]) for the set [parallel]G[parallel] of onedimensional "vectors"

[parallel]G[parallel] = {[+ or -][parallel]a[parallel] : a [member of] G} [subset] R

with vector addition [direct sum] and scalar multiplication [cross product], such that for all r [member of] R and a, b [member of] G,

(G7) [parallel]r [cross product] a[parallel] = [absolute value of (r)] [cross product] [parallel]a[parallel]

(G8) [parallel]a [member of] b[parallel] [less than or equal to] [parallel]a[parallel] [direct sum] [parallel]b[parallel]

Definition 1. Let ABC be a gyrotriangle with sides a, b, c in an Einstein gyrovector space (Vs, [direct sum], [cross product]), and let [h.sub.a], [h.sub.b], [h.sub.c] be three altitudes of ABC drawn from vertices A, B, C perpendicular to their opposite sides a, b, c or their extension, respectively. The number

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is called the gyrotriangle constant of gyrotriangle ABC (here [[gamma].sub.v] = 1/[square root of (1 - [[parallel]v[parallel].sup.2]/[s.sup.2])] is the gamma factor).

(See [2, p. 558].)

Theorem 1. (The Gyrotriangle Constant Principle) Let [A.sub.1]BC and [A.sub.2]BC be two gyrotriangles in a Einstein gyrovector plane ([R.sup.2.sub.s], [product sum], [cross product]), [A.sub.1] [not equal to] [A.sub.2] such that the two gyrosegments [A.sub.1][A.sub.2] and BC, or their extensions, intersect at a point P [member of] [R.sup.2.sub.s]. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(See [2, p. 563].)

Theorem 2. (The Hyperbolic Theorem of Menelaus in Einstein Gyrovector Space)

Let [a.sub.1], [a.sub.2], and [a.sub.3] be three non-gyrocollinear points in an Einstein gyrovector space ([V.sub.s], [product sum], [cross product]). If a gyroline meets the sides of gyrotriangle [a.sub.1][a.sub.2][a.sub.3] at points [a.sub.12], [a.sub.13], [a.sub.23], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(See [2, p. 463].)

Theorem 3. (The Gyrotriangle Bisector Theorem)

Let ABC be a gyrotriangle in an Einstein gyrovector space ([V.sub.s], [product sum], [cross product]), and let P be a point lying on side BC of the gyrotriangle such that AP is a bisector of gyroangle [??]BAC. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(See [3, p. 150].) For further details we refer to the recent book of A. Ungar [2].

2 Main results

In this section, we prove Van Aubel's theorem in hyperbolic geometry.

Theorem 4. If the point P does lie on any side of the hyperbolic triangle ABC, and BC meets AP in D, CA meets BP in E, and AB meets CP in F, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. If we use the Menelaus's theorem in the h-triangles ADC and ABD for the h-lines BPE, and CPF respectively, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

From (1) and (2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the conclusion follows.

Corollary 1. Let G be the centroid of the hyperbolic triangle ABC, and D, E, F are the midpoints of hyperbolic sides BC, CA, and AC respectively. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Proof. If we use theorem 4 for the triangle ABC and the centroid G, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the conclusion follows.

Corollary 2. Let I be the incenter of the hyperbolic triangle ABC, and let the angle bisectors be AD, BE, and CF. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Proof. If we use theorem 3 for the triangle ABC, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

If we use theorem 4 for the triangle ABC and the incenter I, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

From (5) and (6), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the conclusion follows.

The Einstein relativistic velocity model is another model of hyperbolic geometry. Many of the theorems of Euclidean geometry are relatively similar form in the Einstein relativistic velocity model, Aubel's theorem for gyrotriangle is an example in this respect. In the Euclidean limit of large s, s [right arrow] [infinity], gamma factor [[gamma].sub.v] reduces to 1, so that the gyroequality (1) reduces to the

[absolute value of (AP)]/[absolute value of (PD)] = [absolute value of (BC)]/2 [[absolute value of (AE)]/[absolute value of (EC)] x 1/[absolute value of (BD)] + [absolute value of (FA)]/[absolute value of (FB)] x 1/[absolute value of (CD)]

in Euclidean geometry. We observe that the previous equality is a equivalent form to the Van Aubel's theorem of euclidian geometry.

Submitted on October 25, 2011 / Accepted on October 28, 2011

References

[1.] Barbu C. Fundamental Theorems of Triangle Geometry, Ed. Unique, Bacau, 2008 (in Romanian).

[2.] Ungar A.A. Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity, Hackensack, World Scientific Publishing Co. Pte. Ltd., 2008.

[3.] Ungar A.A. A Gyrovector Space Approach to Hyperbolic Geometry, Morgan & Claypool Publishers, 2009.

[4.] Ungar A.A. Analytic Hyperbolic Geometry Mathematical Foundations and Applications, Hackensack, World Scientific Publishing Co. Pte. Ltd., 2005.

Catalin Barbu

"Vasile Alecsandri" National College--Bacau, str. Vasile Alecsandri, nr. 37, 600011, Bacau, Romania

E-mail: kafka_mate@yahoo.com

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. Hyperbolic Geometry includes similar concepts as distance and angle. Both these geometries have many results in common but many are different. There are known many models for Hyperbolic Geometry, such as: Poincare disc model, Poincare half-plane, Klein model, Einstein relativistic velocity model, etc. Here, in this study, we give hyperbolic version of Van Aubel theorem. The well-known Van Aubel theorem states that if ABC is a triangle and AD, BE, CF are concurrent cevians at P, then AP/PD = AE/EC + AF/FB (see [1, p. 271]).

Let D denote the complex unit disc in complex z - plane, i.e.

D = {z [member of] C : [absolute value of (z)] < 1}.

The most general Mobius transformation of D is

z [right arrow] [e.sup.i[theta]] [z.sub.0] + z/1 + [bar.[z.sub.0]]z = [e.sup.i[theta]]([z.sub.0] [direct sum] z),

which induces the Mobius addition [direct sum] in D, allowing the Mobius transformation of the disc to be viewed as a Mobius left gyrotranslation

z [right arrow] [z.sub.0] [direct sum] z = [z.sub.0] + z/1 + [bar.[z.sub.0]]z

followed by a rotation. Here [theta] [member of] R is a real number, z, [z.sub.0] [member of] D, and [bar.[z.sub.0]] is the complex conjugate of [z.sub.0]. Let Aut(D, [direct sum]) be the automorphism group of the grupoid (D, [direct sum]). If we define

gyr : D x D [right arrow] Aut(D, [direct sum]), gyr[a, b] = a [direct sum] b/b [direct sum] a = 1 + a[bar.b]/1 + [bar.a]b,

then is true gyrocommutative law

a [direct sum] b = gyr[a, b](b [direct sum] a).

A gyrovector space (G, [direct sum], [direct sum]) is a gyrocommutative gyrogroup (G, [direct sum]) that obeys the following axioms:

1. gyr[u, v]a x gyr[u, v]b = a x b for all points a, b, u, v [member of] G.

2. G admits a scalar multiplication, [direct sum], possessing the following properties. For all real numbers r, [r.sub.1], [r.sub.2] [member of] R and all points a [member of] G:

(G1) 1 [cross product] a = a

(G2) ([r.sub.1] + [r.sub.2]) [cross product] a = [r.sub.1] [cross product] a [member of] [r.sub.2] [cross product] a

(G3) ([r.sub.1] [r.sub.2]) [cross product] a = [r.sub.1] [cross product] ([r.sub.2] [cross product] a)

(G4) [absolute value of (r)] [cross product] a/[parallel]r [cross product] a[parallel] = a/[parallel]a[parallel]

(G5) gyr[u, v](r [cross product] a) = r [cross product] gyr[u, v]a

(G6) gyr[[r.sub.1] [cross product] v, [r.sub.1] [cross product] v] =1

3. Real vector space structure ([parallel]G[parallel], [direct sum], [cross product]) for the set [parallel]G[parallel] of onedimensional "vectors"

[parallel]G[parallel] = {[+ or -][parallel]a[parallel] : a [member of] G} [subset] R

with vector addition [direct sum] and scalar multiplication [cross product], such that for all r [member of] R and a, b [member of] G,

(G7) [parallel]r [cross product] a[parallel] = [absolute value of (r)] [cross product] [parallel]a[parallel]

(G8) [parallel]a [member of] b[parallel] [less than or equal to] [parallel]a[parallel] [direct sum] [parallel]b[parallel]

Definition 1. Let ABC be a gyrotriangle with sides a, b, c in an Einstein gyrovector space (Vs, [direct sum], [cross product]), and let [h.sub.a], [h.sub.b], [h.sub.c] be three altitudes of ABC drawn from vertices A, B, C perpendicular to their opposite sides a, b, c or their extension, respectively. The number

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is called the gyrotriangle constant of gyrotriangle ABC (here [[gamma].sub.v] = 1/[square root of (1 - [[parallel]v[parallel].sup.2]/[s.sup.2])] is the gamma factor).

(See [2, p. 558].)

Theorem 1. (The Gyrotriangle Constant Principle) Let [A.sub.1]BC and [A.sub.2]BC be two gyrotriangles in a Einstein gyrovector plane ([R.sup.2.sub.s], [product sum], [cross product]), [A.sub.1] [not equal to] [A.sub.2] such that the two gyrosegments [A.sub.1][A.sub.2] and BC, or their extensions, intersect at a point P [member of] [R.sup.2.sub.s]. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(See [2, p. 563].)

Theorem 2. (The Hyperbolic Theorem of Menelaus in Einstein Gyrovector Space)

Let [a.sub.1], [a.sub.2], and [a.sub.3] be three non-gyrocollinear points in an Einstein gyrovector space ([V.sub.s], [product sum], [cross product]). If a gyroline meets the sides of gyrotriangle [a.sub.1][a.sub.2][a.sub.3] at points [a.sub.12], [a.sub.13], [a.sub.23], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(See [2, p. 463].)

Theorem 3. (The Gyrotriangle Bisector Theorem)

Let ABC be a gyrotriangle in an Einstein gyrovector space ([V.sub.s], [product sum], [cross product]), and let P be a point lying on side BC of the gyrotriangle such that AP is a bisector of gyroangle [??]BAC. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(See [3, p. 150].) For further details we refer to the recent book of A. Ungar [2].

2 Main results

In this section, we prove Van Aubel's theorem in hyperbolic geometry.

Theorem 4. If the point P does lie on any side of the hyperbolic triangle ABC, and BC meets AP in D, CA meets BP in E, and AB meets CP in F, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. If we use the Menelaus's theorem in the h-triangles ADC and ABD for the h-lines BPE, and CPF respectively, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

From (1) and (2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the conclusion follows.

Corollary 1. Let G be the centroid of the hyperbolic triangle ABC, and D, E, F are the midpoints of hyperbolic sides BC, CA, and AC respectively. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Proof. If we use theorem 4 for the triangle ABC and the centroid G, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the conclusion follows.

Corollary 2. Let I be the incenter of the hyperbolic triangle ABC, and let the angle bisectors be AD, BE, and CF. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Proof. If we use theorem 3 for the triangle ABC, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

If we use theorem 4 for the triangle ABC and the incenter I, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

From (5) and (6), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the conclusion follows.

The Einstein relativistic velocity model is another model of hyperbolic geometry. Many of the theorems of Euclidean geometry are relatively similar form in the Einstein relativistic velocity model, Aubel's theorem for gyrotriangle is an example in this respect. In the Euclidean limit of large s, s [right arrow] [infinity], gamma factor [[gamma].sub.v] reduces to 1, so that the gyroequality (1) reduces to the

[absolute value of (AP)]/[absolute value of (PD)] = [absolute value of (BC)]/2 [[absolute value of (AE)]/[absolute value of (EC)] x 1/[absolute value of (BD)] + [absolute value of (FA)]/[absolute value of (FB)] x 1/[absolute value of (CD)]

in Euclidean geometry. We observe that the previous equality is a equivalent form to the Van Aubel's theorem of euclidian geometry.

Submitted on October 25, 2011 / Accepted on October 28, 2011

References

[1.] Barbu C. Fundamental Theorems of Triangle Geometry, Ed. Unique, Bacau, 2008 (in Romanian).

[2.] Ungar A.A. Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity, Hackensack, World Scientific Publishing Co. Pte. Ltd., 2008.

[3.] Ungar A.A. A Gyrovector Space Approach to Hyperbolic Geometry, Morgan & Claypool Publishers, 2009.

[4.] Ungar A.A. Analytic Hyperbolic Geometry Mathematical Foundations and Applications, Hackensack, World Scientific Publishing Co. Pte. Ltd., 2005.

Catalin Barbu

"Vasile Alecsandri" National College--Bacau, str. Vasile Alecsandri, nr. 37, 600011, Bacau, Romania

E-mail: kafka_mate@yahoo.com

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Author: | Barbu, Catalin |
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Publication: | Progress in Physics |

Date: | Jan 1, 2012 |

Words: | 1396 |

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