Golden Sections of Triangle Centers in the Golden Triangles

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1 Forum Geometricorum Volume 16 (016) FRUM GEM ISSN Golden Sections of Tringle Centers in the Golden Tringles Emmnuel ntonio José Grcí nd Pul Yiu bstrct. golden tringle is one whose vertices re mong the vertices of regulr pentgon. There re two kinds of golden tringles, short nd tll, which re isosceles tringles with verticl ngles 108 nd 36 respectively. We consider some bsic tringle centers of short nd tll golden tringles shring one vertex nd with the sme circumcircle, nd exhibit pirs of bsic tringle centers divided in the golden rtio by nother tringle center. s is well known, the golden rtio nturlly occurs in the regulr pentgon, s the rtio of the length of digonl d nd side : ϕ := d = 1 ( 5+1). The intersection of two digonls divides ech in the golden rtio. If BCDE is regulr pentgon, nd the digonls D nd BE intersect t P (see Figure 1), then BE BP = BP PE = ϕ, D DP = DP P = ϕ. For lter use, we note the following simple trigonometric rtios from Figure 1: cos 36 = d/ = ϕ, sin 18 = / d = 1 ϕ. B d P d E d d d C D Figure 1. The golden section Figure. Short nd tll golden tringles Given regulr pentgon, the subtringles with vertices mong those of the pentgon re ll isosceles. They fll into two types: (i) those with three djcent vertices of the pentgon hve ngles 108, 36, 36, which we cll short golden tringles, Publiction Dte: pril 1, 016. Communicting Editor: Nikolos Dergides.

2 10 E.. J. Grcí nd P. Yiu (ii) those with only two djcent vertices of the pentgon hve ngles 36, 7, 7, which we cll tll golden tringles. In this note we consider golden sections in the two kinds of golden tringles. For purpose of comprison, we consider pir of short nd tll golden tringles inscribed in the sme regulr pentgon (see Figure ). The short golden tringle T s := hs sides d,, ; the tll golden tringle T t := hs sides, d, d. They shre the sme circumcenter. Denote by R their common circumrdius. Note tht the res Δ i, i =s, t, of the golden tringles re in the golden rtio: Δ t Δ s = 1 d sin 7 1 sin 108 = d = ϕ. For i =s, t, since the golden tringle T i is isosceles, its tringle centers re ll on the (common) perpendiculr bisector of the side B i C i. We shll cll this the center line of the golden tringles; It contins the midpoints F i of B i C i (see Figure 3). H s P I t H t Figure 3 Here re some simple constructions of the bsic tringle centers of T s nd T t. (1) The incenter of T s is the intersection of the center line with the perpendiculr of t ; it is lso the reflection of in the side. From this, the inrdius of T s is r s = = = R cos 7 = R ϕ.

3 Golden sections of tringle centers in the golden tringles 11 () The incenter I t of T t is the intersection of the digonls nd.it is lso the reflection of in the side. From this, the inrdius of T t is r t = tn 36 = R sin 36 tn 36 = R sin 36 = R 1 ( ϕ ϕ ) cos 36 = R 4 ϕ ϕ = R (3ϕ 4). (3) Let H s be the orthocenter of T s. Clerly H s = =7. Since H s is the isogonl conjugte of in T s, H s = = 36 =7. Therefore, tringle H s is (tll) golden tringle, nd H s = d = ϕ = H s = ϕr. lso, by the ngle bisector theorem, H s = H s = ϕ. s s This shows tht divides H s in the golden rtio. Since bisects ngle H s, the sme resoning shows tht divides H s in the golden rtio. (4) In the tll golden tringle T t, the orthocenter H t is the intersection of the center line with the perpendiculr to t. Note tht H t =Rcos 7 = R / d = R ϕ. Since H t is the isogonl conjugte of in T t, by the ngle bisector theorem, I t = = ϕ. I t H t H t Therefore, I t divides H t in the golden rtio (5) Since nd I t re the reflections of nd in, I t =, nd = = ϕ. I t Therefore, divides I t in the golden rtio. (6) In the tll golden tringle, divides H t in the golden rtio. H t R cos 36 = =cos36 = ϕ. R We summrize these results in the following proposition. Proposition 1. Let T i, i =s, t be golden tringles shring common vertex nd the sme circumcircle with center. Let H i nd I i denote the orthocenter nd incenter of T i. () The incenter I i divides H i or H i in the golden rtio, ccording s i = s, t. (b) The circumcenter divides ech of I t nd H t in the golden rtio. (c) divides H s in the golden rtio.

4 1 E.. J. Grcí nd P. Yiu Some observtions by Nikolos Dergides: (i) is the midpoint of H t. (ii) If is the ntipode of on the circumcircle, the tringles H s nd re similr to T s, nd since = = R, wehve H s = = Rϕ. (iii) The tringle I t H s hs right ngle t, nd since H s =, is the midpoint of H s I t. (iv) The segments = R ϕ, = R, H s = Rϕ re in geometric progression (with common rtio ϕ). Since ϕ =1+ 1 ϕ,wehveh s = +. This mens tht the circles (H s ), (H s ) nd ( ) re concurrent t point D which lies on the line (see Figure 4). H s P I t D H t Figure 4 For i =s, t, the incircle of T i is tngent to the side B i C i t its midpoint F i. Since this midpoint lso lies on the nine-point circle of T i, it is the Feuerbch point of T i. The nine-point circle of T i, i =s, t, lso contins the midpoints M i,b. M i,c of the sides C i nd B i. Proposition. () divides in the golden rtio. (b) The incenter I t divides in the golden rtio. Proof. () Let P be the intersection of the digonls nd (see Figure 3). Since nd re prllel, = P = ϕ. Therefore, divides in the golden rtio. (b) Since I t is the intersection of the digonls nd, FsIt I t = BsCs = ϕ.

5 Golden sections of tringle centers in the golden tringles 13 F s N s N s M s,c M s,b F t M t,c M t,b N t N t Figure 5 Figure 6 Proposition 3. () For the short golden tringle T s with nine-point center N s, the incenter divides N s in the golden rtio. (b) For the tll golden tringle T t, the nine-point center N t divides in the golden rtio (See Figure 5). Proof. () The inrdius of T s is r s = R FsNs ϕ. Therefore, = R rs = ϕ, nd divides N s in the golden rtio. (b) = R cos 36 = R ϕ. Therefore, N t = ϕ, nd N t divides in the golden rtio. Proposition 4. For {i, j} = {s, t}, the nine-point center N i of T i is the reflection of F j in the center. Proof. () Since is the reflection of in, = = r s = R ϕ, N s = + N s = R ϕ + R = R ( ) 1 ϕ +1 = R ϕ = R cos 36 =. Therefore, N s is the reflection of in. (b) Since N t = N s, N t = N s + = + = is the reflection of in. Therefore, the nine-point center N s is the ntipode of on the circle, center, pssing through, which is the inscribed circle of the regulr pentgon. It follows tht N s M s,b nd N s M s,b re right ngles. This mens tht M s,b nd M s,c re tngents to the nine-point circle of T s t M s,b nd M s,c respectively (see Figure 6). The line M s,b psses through M t,b, which divides M s,b in the

6 14 E.. J. Grcí nd P. Yiu golden rtio. Similrly, M s,c is the tngent t M s,c nd is divided in the golden rtio by M t,c. The sme resoning lso leds to the following. (i) The points M s,b,, M t,c re colliner, nd divides M s,b M t,c in the golden rtio. Furthermore, the line contining them is tngent to the nine-point circle of T t t M t,c. (ii) The points M s,c,, M t,b re colliner, nd divides M s,c M t,b in the golden rtio. Furthermore, the line contining them is tngent to the nine-point circle of T t t M t,b. We conclude this note with few more division in the golden rtio with points in Figure 5. The simple proofs re omitted. For i =s, t, let F i be the ntipode of F i on the nine-point circle of T i. Then () divides N s in the golden rtio, (b) divides ech of the segments N t nd N s in the golden rtio, (c) divides F t in the golden rtio, (d) divides F t in the golden rtio. Sttement (d) follows from Proposition (b) nd trnsltion by R long the center line. Figure 7 summrizes the golden sections in this note, ech indicted by longer solid segment followed by shorter dotted segment. The endpoints nd the division points re indicted on the center line. H s F s N s F t N I t H t References Figure 7 [1] M. Btille, nother simple construction of the golden section, Forum Geom., 11 (011) 55. [] K. Hofstetter, simple construction of the golden section, Forum Geom., (00) [3] J. Niemeyer, simple construction of the golden section, Forum Geom., 11 (011) 53. Emmnuel ntonio José Grcí: CEDI Bilingul School, Cmil Henríquez Ureñ, 0, Snto Domingo, Dominicn Republic. E-mil ddress: emmnuelgeogrci@gmil.com Pul Yiu: Deprtment of Mthemticl Sciences, Florid tlntic University, 777 Gldes Rod, Boc Rton, Florid , US E-mil ddress: yiu@fu.edu

Andrew Ryba Math Intel Research Final Paper 6/7/09 (revision 6/17/09)

Andrew Ryba Math Intel Research Final Paper 6/7/09 (revision 6/17/09) Andrew Ryb Mth ntel Reserch Finl Pper 6/7/09 (revision 6/17/09) Euler's formul tells us tht for every tringle, the squre of the distnce between its circumcenter nd incenter is R 2-2rR, where R is the circumrdius