DEMONSTRATIO MATHEMATICA Vol. XLII No 2 2009

Nilgun Sônmez

SOME FEATURES IN CONNECTION WITH TRIANGLES IN THE POINCARÉ UPPER HALF PLANE

Abstract. In this work, it is shown that the product of lengths that are segments reserved to altitudes of orthocenter is constant in the Poincaré triangle. I defined to Carnot theorem Poincaré upper half plane.

1. Introduction The Poincaré upper half plane geometry has been introduced by Henri Poincaré. Poincaré upper half plane is the upper half plane of the Euclidean analytical plane M2. Although the points in the Poincaré upper half plane are the same as the points in the upper half plane of the Euclidean analytical plane R2, the lines and the distance function between any two points are different. The lines in the Poincaré upper half plane are defined by

2 aL = {(x, y) € M | x = a, y > 0, a € M, a constant} half lines and

2 2 2 2 c:L r = {(x, y) G M I (x - c) + y = r , y > 0, c, r G R, c, r constant, r > 0} half circles.

If A={x\, y\) and B—(x2, 2/2 ) are any two points in H then the Poincaré distance between these points is given by

|ln(y2/yi)|, if X\ = X2

dH(A,B) = i lnfy2{xi-c + r)\ In if Xl ± X2 \yi(x2 - c + r)J '

1991 Mathematics Subject Classification: 51K05, 51K99. Key words and phrases: Poincaré upper half plane, Poincaré circle, Poincaré triangle. 404 N. Sònmez where c={yl~y\ + xl- X\)/2{X2 - xi),

2 2 r = ^(xi - c) + y\ = yj(x2 ~ c) + y\. The Poincaré upper half plane geometry is non-Euclidean, since it fails to satisfy the parallel postulate but satisfies all the remaining twelve axioms of the Euclidean plane geometry [2, 3, 4]. In this Poincaré upper half plane geometry, the lines and the function of distance are different, therefore, it seems interesting to study the Poincaré analogues of the topics that include the concept of distance in the Euclidean geometry. A few of such topics have been studied by some authors [1, 3-9]. In this work, it is shown that the product of lengths that are segments reserved to altitudes of orthocenter is constant in the Poincaré triangle. I defined to Carnot theorem Poincaré upper half plane. THEOREM 1. Every Euclidean circle in the upper half-plane is also a hy- perbolic circle [5, p. 80].

LEMMA 1. In the hyperbolic triangle ABC let a,/3,j angels denote at A, B, C and a, b, c denote the hyperbolic lengths of the sides opposite A, B, C, respectively, then sin a sin/? sin7 —-— = —— = —-— (bines lheorem) 15, p. 125 . sinh a sinh b sinh c

LEMMA 2 (Pythagoras Theorem). Let ABC be a hyperbolic triangle with a right angle at C. If a, b, c, are the hyperbolic lengths of the sides opposite A, B, C, respectively, then cosh c = cosh a cosh b.

LEMMA 3. The Poincaré upper half plane H is a neutral geometry [3, p. 129]. LEMMA 4. In a neutral geometry the center and radius of a circle are determined by any three points on the circle [3, p. 152]. LEMMA 5. For any circle in a neutral geometry, the perpendicular bisector of any chord contains the center [3, p. 153]. We know that three altitudes of a triangle intersect at one point in the Euclidean geometry that is neutral geometry. Now, let us examine the va- lidity of this statement in the Poincaré upper half plane. THEOREM 2. The three altitudes of a hyperbolic triangle intersect in a common point in the Poincaré upper half plane.

Proof. Let ABC be a triangle and let's draw a triangle A'B'C' that is the inverse medial triangle of triangle ABC. In other words, namely, triangle Triangles in the Poincaré upper half plane 405

A

C •B'

A Fig. 1.

ABC is formed by joining the midpoints of the sides of the triangle A'B'C' (see Figure 1) and these midpoints are the corners of the triangle ABC. In Euclidean geometry that is neutral, there exists a unique circle passing through three vertices of the triangle A'B'C' (see Lemma 4) and a center of this circle is the point of intersection of all altitudes of the triangle ABC (i.e. orthocenter of ABC). Since Poincaré upper half plane is a neutral geometry (see Lemma 3) there exists a unique circle passing through three vertics of the triangle A'B'C' (see Lemma 4) and the center of this circle is the intersection point of the perpendicular bisectors of the triangle A'B'C' (see Lemma 5). Consequently, all altitudes of a hyperbolic triangle intersect in a common point in the Poincaré upper half plane. •

THEOREM 3. The product of lengths that are segments reserved to altitudes of orthocenter is constant in the Poincaré triangle.

Proof. Let ABC be a triangle that satisfies Theorem 2. (See Figure 2)

A

B C Fig. 2.

If we use the sines theorem in the triangles FHB, EHC, EH A, respec- tively (see Lemma 1), then 406 N. Sonmez

sin 90 sin a sinh BH = sinh FH' (1) sinh FH = sin a sinh BH, sin 90 sin /3 sinh HC = sinh HE'

(2) sinh^C=Sinh^ sin/3 sin 90 sin 9 sinhAff sinh HE' . „ sinh HE 3 sinfl = . sinh AH Because of EH A = DHB (ASA), if we use the sines theorem in the triangle DHB, then sin 90 sin 0 sinh BH = sinh HD* (4) sinh HD 6 = sin 6 sinh BH. Because of FHB S EHC (ASA), a = 0

sinh FH = ^^ sinh BH sinh HC (5) sinh FH sinh HC = sinh HE sinh BH. If we put the value of sin 6 = AH ^^e equation (3) in the equation (4) then , „„ sinh HE , „„ sinh HD = ——— sinh BH, sinh AH (6) sinh HD sinh AH = sinh HE sinh BH. We find following equality from (5) and (6). sinh FH sinh HC = sinh BH sinh HE = sinh AH sinh HD. Furthermore, we can find three equalities. Let us employ the sines theorem in the triangles ACF, ABE, respectively. sin (3 sin 90 sinh AF ~ sinh AC' (7). . sin/. 03 = sinh AF sinh AC' sin (3 sin 90 sinhAE sinh AB' (8) sinh AE = sin /3 sinh AB. Triangles in the Poincaré upper half plane 407

If we put the value of sin/3 of the equation (7) in the equation (8) then

sinh AE = SÌnj1 ^^ sinh AB, sinh AE sinh AC = sinh AF sinh AB. smh AC We see FAH = DCH because of HDC ^ HFA (ASA). Let us employ the sines theorem in the triangles ADC, BCE, respectively sin 9 _ sin 90 sinh CD = sinh CA' • sinh CD (9) sing n = . v ' sinh CA sin 9 sin 90 sinh CE sinh CB' (10) sinh CE = sin 9 sinh CB. If we put the value of sin# of the equation (9) in the equation (10) then . , sinh CD . , sinh CE = . , ^ , sinh CB, sinh CM sinh CE sinh CA = sinh CD sinh CB. Let us employ the sines theorem in the triangles ABD, CBF, respectively sin £ sin 90 sinh BD sinh BA' sinh BD (11) sin£ = sinh BA' sin £ sin 90 sinh BF sinh BC' (12) sinh BF = sin £ sinh BC. If we put the value of sin£ of the equation (11) in the equation (12) then . , „„ sinh BD . , _ sinh BF = ——— sinh BC, sinh BA sinh BF sinh BA = sinh BD sinh BC. m

THEOREM 4 (Carnot Theorem). Let ABC be an arbitrary triangle in the upper half-plane, H — its hyperbolic orthocenter (see Theorem 2), altitudes AH, BH, CH intersect the opposite sides in points D, E, F, respectively (see Figure 2). There is the following correlation between lengths of two triples of disjoint segments cosh AF cosh BD cosh CE = cosh FB cosh DC cosh EA. 408 N. Sonmez

Proof. If we employ the Pythagoras theorem on the triangles HBD, HCD, HCE, HAE, HAF, HBF, respectively (see Lemma 2),

cosh HB = cosh BD cosh DH, (13) cosh HC = cosh DH cosh DC, cosh HC = cosh HE cosh CE, (14) cosh HA = cosh EA cosh HE, cosh HA = cosh AF cosh FH, (15) cosh HB = cosh FB cosh FH.

If we take ratios the equations (13), (14) and (15) among themselves, respec- tively then cosh HB _ cosh BD cosh HC ~ cosh DC" cosh HC _ cosh CE cosh HA cosh EA' cosh HA _ cosh^4F cosh HB ~~ cosh FB' cosh HA cosh HB cosh HC ^ cosh HB cosh HC cosh HA ~~ ' cosh AF cosh BD cosh CE _ cosh FB cosh DC cosh EA ~ ' cosh AF cosh BD cosh CE = cosh FB cosh DC cosh EA. •

References

[1] J. W. Anderson, Hyperbolic Geometry, Springer, London, Berlin, Heidelberg, 1999. [2] E. F. Krause, Taxicab Geometry, Addison-Wesley, Menlo Park, California, 1975. [3] R. S. Millman, G. D. Parker, Geometry a metric approach with models, Verlag New York Inc., 1991, p. 367. [4] N. Sönmez, On the geometry of Poincare half plane, Ph. D. Thesis, Osmangazi Univer- sity, Institute of Science and Technology, 2006, p. 98. [5] S. Stahl, The Poincare half plane a gateway to modern geometry, Jones and Barlett Publishers, Boston, 1993 p. 298. [6] S. Stahl, Geometry from Euclid to Knots, Pearson Education, Inc., Upper Saddle Riveri New Jersey, 2002, p. 457. [7] J. Stilwell, Sources of hyperbolic geometry, Amer. Math. Soc. (1996), p. 153. Triangles in the Poincaré upper half plane 409

[8] H. H. Ugurlu, M. Kazaz, A. ve Özdemir, Stewart, Ceva and Menelaus Theorems for geodesic triangles on the dual hyperbolic unit sphere of the dual Lorentzian 3-space D\, III. Geometry Symposium, Eskisehir, 2005. [9] A. T. Yakut, Vertex angles of simplices in hyperbolic space, Ph. D. Thesis, Gazi Univer- sity Institute of Science and Tecnology, 2004, p. 142.

AFYON KOCATEPE UNIVERSITY FACULTY OF SCIENCE AND LITERATURES DEPARTMENT OF MATHEMATICS ANS CAMPUS, 03200 - AFYONKARAHISAR, TURKEY E-mail: [email protected]

Received April 2, 2008; revised version November 28, 2008.