Equal Area Triangles Inscribed in a Triangle

Transcription

1 INTERNATIONAL JOURNAL OF GEOMETRY Vol , No 1, Equl Are Tringles Inscribed in Tringle SÁNDOR NAGYDOBAI KISS Abstrct We exmine the condition tht two or more inscribed tringles in tringle hve sme re, nd the reltions between the equl re tringles nd the isotomic points 1 Introduction A tringle XY Z is sid to be inscribed in tringle ABC if X lies on BC, Y lies on CA, nd Z lies on AB In mthemtics literture there re not mny references bout the inscribed tringles in tringle Personlly I hve found two documents [1], [2] in which equilterl tringles inscribed in tringle re discussed In this pper, in section 1 we exmine the condition tht two or more inscribed tringles hve the sme re nd in section 2 we enounce theorem, which permits the construction of inscribed equl re tringles in tringle In section 3 we exmine the reltions between the equl re tringles nd the isotomic points Finlly we give some properties of the inscribed equl re tringles We use the brycentric coordintes with respect to the tringle ABC: A 1, 0, 0, B 0, 1, 0, C 0, 0, 1 Consider the points X, Y, Z with bsolute brycentric coordintes: X 0, u, 1 u, Y 1 v, 0, v, Z w, 1 w, 0 Let s denote the signed re of the A 1 A 2 A n polygon with the symbol σ[a 1 A 2 A n ] nd the re of the tringle ABC with Let X Y Z be nother inscribed tringle in the tringle ABC: X 0, u, 1 u, Y 1 v, 0, v, Z w, 1 w, Mthemtics Subject Clssifiction 97G40 Key words nd phrses Brycentric coordintes, isotomic points Received: In revised form: Accepted:

2 20 Sándor Ngydobi Kiss 2 The conditions under which two inscribed tringles hve equl re The re of the tringles XY Z nd X Y Z re: 0 u 1 u σ[xy Z] 1 v 0 v [uvw + 1 u1 v1 w], w 1 w 0 σ[x Y Z 0 u 1 u ] 1 v 0 v w 1 w 0 [u v w + 1 u 1 v 1 w ] The tringles XY Z nd X Y Z hve n equl re if uvw + 1 u1 v1 w u v w + 1 u 1 v 1 w 21 If we consider fix the tringle X Y Z nd the pir of vribles v, w, then from condition 21 we obtin two vlues for the vrible u This signifies tht there re n infinite number of inscribed tringles which hve the sme re s the fixed X Y Z tringle Severl exmples follow: Let A, B, C be the midpoint of the side BC, CA, AB, nd X A 0, 1 2, 1 2, Y B 1 2, 0, 1 2, Z C w 1 2 nd σ[x Y Z ] σ[a B C ] 4 1 2, 1 2, 0 Following u v 21 If v 2 3 nd w 1, then u 0 or u 6 In this cse X 0, 0, C or X 0, 6, 5 nd Y 3, 0, 2 1, Z 3 4, 3 4, 0 It cn esily verified tht σ[xy Z] 4 σ[x Y Z ] nd σ[x Y Z] 4 σ[x Y Z ] 22 If v 2 nd w 1, then u 1 8 or u 1 In this cse 8 X 1 0, 1 8, 7 or X1 8 0, 1 8, 9, Y 1 1, 0, 2, Z 1 1, 0, 0 A, 8 σ[x 1 Y 1 Z 1 ] 4 σ[x Y Z ] nd σ[x 1 Y 1Z 1 ] 4 σ[x Y Z ] 23 If v 4 nd w 2, then u 7 or u 19 In this cse 4 12 X 2 0, 7 4, 11 or X2 4 0, 19 12, 31, Y 2 5, 0, 4, Z 2 2, 1, 0, 12 σ[x 2 Y 2 Z 2 ] 4 σ[x Y Z ] nd σ[x 2 Y 2Z 2 ] 4 σ[x Y Z ] Now we represent the tringles XY Z, X Y Z nd X1 Y 1Z 1 Figure 1 If the point X is on the line BC nd t BX, then the bsolute brycentric XC coordintes of X cn be obtined s convex combintion of B nd C: X 1 t B + t t + 1 t + 1 C 1 t + 1 B + t C nd BX t t + 1 t + 1 BC,

3 Equl Are Tringles Inscribed in Tringle 21 XC 1 t + 1 BC This result is used lso in the cse of XY Z nd X 1 Y 1Z 1 tringles: Y 2 3 C A, Z 1 4 A B, CY Y A 1 2, CY 1 3 CA, Y A 2 3 CA, AZ ZB 3, AZ 3 4 AB, ZB 1 4 AB, X B C, BX 1 X 1 C 9, BX BC, X 1C 1 8 BC, Y 1 2C A, B Z C' CY 1 Y 1 A 1 2, CY 1 CA, Y 1 A 2CA A_ Z 1 A' B' Y _ X C * X 1 Figure 1 Y 1 3 Construction of the inscribed equl re tringles Theorem 31 If X, Y, Z is the symmetric of X, Y, Z with respect to A, B, C, then σ[xy Z] 1 u v w + vw + wu + uv σ[x Y Z ], 31 σ[x Y Z] u + vw wu uv σ[xy Z ], 32 σ[xy Z] v vw + wu uv σ[x Y Z ], 33 σ[xy Z ] w vw wu + uv σ[x Y Z] 34 Proof In this cse X 0, 1 u, u, Y v, 0, 1 v, Z 1 w, w, 0 Figure 2 Consequently, σ[x Y Z 0 1 u u ] v 0 1 v [uvw+1 u1 v1 w] σ[xy Z], 1 w w 0 σ[x 0 1 u u 0 u 1 u Y Z] 1 v 0 v v 0 1 v w 1 w 0 1 w w 0 σ[xy Z ]

4 22 Sándor Ngydobi Kiss B Z' Z C' A Y' X A' X' B' Y C Figure 2 31 Consider the following numericl exmples: X 0, 0, 1 C, Y 1 3, 0, 2 1, Z 3 4, 3 2 4, 0 Therefore X 0, 1, 0 B, Y 3, 0, 1, 3 3 Z 4, 1 4, 0 Figure 3 Z B _ X' C' Z' A A' Figure 3 Y' B' Y X _ C Now we clculte elementry the re of the tringles XY Z, X Y Z, X Y Z, XY Z : σ[xy Z] σ[azy ] σ[bcz] , σ[x Y Z ] σ[az Y ] σ[bcy ] , 2b 3 b 3 3c 4 sin A 1 2 c 4 sin B c 4 sin A 1 2 2b 3 sin C

5 Equl Are Tringles Inscribed in Tringle 23 σ[x Y Z] σ[azy ] σ[bcy ] , 2b 3 3c 4 sin A 1 2 b 3 sin C σ[xy Z ] σ[az Y ] σ[bcz ] b 3 c 4 sin A 1 2 3c 4 sin B Similrly we obtin: σ[xy Z] σ[x Y Z ] 2 nd σ[xy Z ] σ[x Y Z] Let, b, c be the sidelength of tringle ABC nd s its semiperimeter: 2s + b + c The touchpoints of the incircle with the side BC, CA, AB re D 0, s c, s b s c, E, 0, s s b, F, s b b c c The A-excircle, B-excircle nd C-excircle touch the side BC, CA, AB in the points D 0, s b, s c s, E, 0, s c, respectively b b s F, s b, 0 Figure 4 c c B F' C' F A I E D A' Figure 4 Corollry 31 The pirs of points D, D, E, E, F, F re symmetric with respect to the midpoints A, B, C of the side BC, CA, AB nd consequently σ[def ] σ[d E F ] D' B' 2s s bs c bc σ[d EF ] σ[de F ] b + c 2 + b 2 + c 2 2bc σ[de F ] σ[d EF ] c + b2 + b 2 + c 2 2c σ[def ] σ[d E F ] + b c2 + b 2 + c 2 2b E' C, 35 36, 37 38

6 24 Sándor Ngydobi Kiss We deduce the equlity 35: σ[def ] σ[d E F ] 1 s c s s b + s s b + s bs c + s cs b c bc c b + c s s s b + s s b + s bs c + s cs b c bc c b s b s s b + s s b + s bs c + s cs b c bc c b s b s c s s b s s b s cs + + b c bc b s s b s s s b s cs + + c b bc b s b s c 1 b + s b + s c bc b s b s c + s b + s c bc b s s b + b c bc 2s s bs c bc 33 The excircles touch the side BC, CA, AB in severl points P 0, s, s, Q s b, 0, s s, R b b c, s c, 0, c P Figure 5 0, s, s s, Q b, 0, s, R s c, s b b c c, 0, Corollry 32 The pirs of points P, P, Q, Q, R, R re symmetric with respect to the midpoints A, B, C of the side BC, CA, AB nd consequently σ[p QR] σ[p Q R ] 2s3 bc + c + bs + bc bc σ[p QR] σ[p Q R ] + b + c2 b 2 c 2 σ[p Q R] σ[p QR ] + b + cb2 c 2 2 σ[p QR ] σ[p Q R] + b + cc2 2 b

7 Equl Are Tringles Inscribed in Tringle 25 R' I c P B Q' C' A Ạ' R B' C P' I b Q I Figure 5 Here we deduce the equlity 310: Figure 5 s σ[p QR] σ[p Q R ] + s s b c s s c s s [bc sb + c ] bc s [2bc b + c + b + c ] 2bc + b + c2 b 2 c 2 s b 4 Inscribed equl re tringles nd the isotomic points Let L, M, N be three points on the sides of the tringle ABC: L BC, M CA, N AB It is well-known tht if the tringles LMN nd ABC re perspectives nd if L, M, N is the symmetric of L, M, N with respect to A, B, C, then the tringles L M N nd ABC re perspectives too nd conversely Note their perspectors with P nd P : P AL BM CN nd P AL BM CN The P nd P points re clled isotomic points Theorem 41 If the brycentric coordintes of the point P re α : β : γ, αβγ 0, then the brycentric coordintes of its isotomic point P re

8 26 Sándor Ngydobi Kiss 1 α : 1 β : 1 nd γ σ[lmn] σ[l MN] σ[lm N] σ[lmn ] 2αβγ β + γγ + αα + β σ[l M N ], 41 αβ 2 + γ 2 β + γγ + αα + β σ[lm N ], 42 βγ 2 + α 2 β + γγ + αα + β σ[l MN ], 43 γα 2 + β 2 β + γγ + αα + β σ[l M N] 44 Proof In this cse the brycentric coordintes of the points L, M, N nd L, M, N re L 0; β : γ, M α : 0 : γ, N α : β : 0, L 0 : γ : β, M γ : 0 : α, N β : α : 0 The bsolute brycentric coordintes of this points re β L 0, β+γ, γ α, M β+γ γ+α, 0, γ, N γ+α L 0, γ β+γ, Consequently β γ, M β+γ γ+α, 0, σ[lmn] 1 β + γγ + αα + β α α+β, α β, N γ+α α+β, 0 β γ α 0 γ α β 0 2αβγ β + γγ + αα + β σ[l M N ], σ[l 1 0 γ β MN] β + γγ + αα + β α 0 γ α β 0 αβ 2 + γ 2 β + γγ + αα + β σ[lm N ] β α+β, 0, α α+β, 0 Henceforth we consider pirs of isotomic points formed with tringles centers [3] Let s denote the pirs of these tringle centers with Xi nd Xj Let s consider L i BC AXi, M i CA BXi, N i AB CXi We will write the formuls for some prticulr cses

9 Equl Are Tringles Inscribed in Tringle The isotomic conjugte of the incenter X1 : b : c is the center X75: σ[l 1 M 1 N 1 ] σ[l 75 M 75 N 75 ] σ[l 75 M 1 N 1 ] σ[l 1 M 75 N 75 ] σ[l 1 M 75 N 1 ] σ[l 75 M 1 N 75 ] σ[l 1 M 1 N 75 ] σ[l 75 M 75 N 1 ] 2bc, 45 b + cc + + b b 2 + c 2, 46 b + cc + + b bc 2 + 2, 47 b + cc + + b c 2 + b 2 48 b + cc + + b 42 The isotomic conjugte of the centroid X2 1 : 1 : 1 is the centroid itself: σ[l 2 M 2 N 2 ] σ[a B C ] The isotomic conjugte of the circumcenter X3 cos A : b cos B : c cos C is the center X264: σ[l 3 M 3 N 3 ] σ[l 264 M 264 N 264 ] 410 2bc cos A cos B cos C b cos B+c cos Cc cos C + cos A cos A+b cos B, σ[l 264 M 3 N 3 ] σ[l 3 M 264 N 264 ] 411 cos Ab 2 cos 2 B + c 2 cos 2 C b cos B+c cos Cc cos C + cos A cos A+b cos B, σ[l 3 M 264 N 3 ] σ[l 264 M 3 N 264 ] 412 b cos Bc 2 cos 2 C + 2 cos 2 A b cos B+c cos Cc cos C + cos A cos A+b cos B, σ[l 3 M 3 N 264 ] σ[l 264 M 264 N 3 ] 413 c cos C 2 cos 2 A + b 2 cos 2 B b cos B+c cos Cc cos C + cos A cos A+b cos B 44 The isotomic conjugte of the orthocenter X4 cos B cos C : b cos C cos A : c cos A cos B is the center X69: σ[l 4 M 4 N 4 ] σ[l 69 M 69 N 69 ] 2 cos A cos B cos C, 414 the re of the orthic tringle σ[l 69 M 4 N 4 ] σ[l 4 M 69 N 69 ] cos Ab2 cos 2 C + c 2 cos 2 B bc σ[l 4 M 69 N 4 ] σ[l 69 M 4 N 69 ] cos Bc2 cos 2 A + 2 cos 2 C c σ[l 4 M 4 N 69 ] σ[l 69 M 69 N 4 ] cos C2 cos 2 B + b 2 cos 2 A b, 415,

10 28 Sándor Ngydobi Kiss 45 The isotomic conjugte of the symmedin point X6 2 : b 2 : c 2 is the 3 rd Brocrd point X76: σ[l 6 M 6 N 6 ] σ[l 76 M 76 N 76 ] σ[l 76 M 6 N 6 ] σ[l 6 M 76 N 76 ] σ[l 6 M 76 N 6 ] σ[l 76 M 6 N 76 ] 2 2 b 2 c 2 b 2 + c 2 c b 2, b 4 + c 4 b 2 + c 2 c b 2, 419 b 2 c b 2 + c 2 c b c b 4 σ[l 6 M 6 N 76 ] σ[l 76 M 76 N 6 ] b 2 + c 2 c b The isotomic conjugte of the Gergonne point X7 is the Ngel point X8 b + c : c + b : + b c see Corollry 31: σ[l 7 M 7 N 7 ] σ[l 8 M 8 N 8 ] σ[def ] σ[d E F ] 422 2s s bs c bc σ[l 8 M 7 N 7 ] σ[l 7 M 8 N 8 ] σ[d EF ] σ[de F ] 423, b + c 2 + b 2 + c 2 2bc σ[l 7 M 8 N 7 ] σ[l 8 M 7 N 8 ] σ[de F ] σ[d EF ] 424 c + b2 + b 2 + c 2 2c σ[l 7 M 7 N 8 ] σ[l 8 M 8 N 7 ] σ[def ] σ[d E F ] b c2 + b 2 + c 2 2b 47 The isotomic conjugte of the Spieker center X10 b + c : c + : + b is the center X86: σ[l 10 M 10 N 10 σ[l 86 M 86 N 86 ] 426,, 2b + cc + + b b + c + 2c + + 2b + b + 2c, σ[l 86 M 10 N 10 ] σ[l 10 M 86 N 86 ] 427 b + cb2 + c b + 2c b + c + 2c + + 2b + b + 2c, σ[l 10 M 86 N 10 ] σ[l 86 M 10 N 86 ] 428 c + c b 2 + 2bc + 2b b + c + 2c + + 2b + b + 2c, σ[l 10 M 10 N 86 ] σ[l 86 M 86 N 10 ] b2 + b 2 + 2c 2 + 2c + 2bc b + c + 2c + + 2b + b + 2c

11 Equl Are Tringles Inscribed in Tringle 29 It is possible to continue this clculus with nother pirs of isotomic points 5 Properties of inscribed tringles in tringle Theorem 51 If X, Y, Z re the symmetrics of X, Y, Z with respect to A, B, C then the sum of the res of the tringles XY Z, X Y Z, XY Z, XY Z is equl with the re of the tringle ABC, ie σ[xy Z] + σ[x Y Z] + σ[xy Z] + σ[xy Z ] σ[abc] 51 σ[x Y Z ] + σ[xy Z ] + σ[x Y Z ] + σ[x Y Z] σ[abc] 52 Proof From the formul 31 we obtin: σ[xy Z] + u + v + w vw wu uv σ[abc] Summing equlity 32, 33 nd 34 we obtin: σ[x Y Z] + σ[xy Z] + σ[xy Z ] u + v + w vw wu uv We verify the equlity 51 for the tringles from 31: σ[xy Z]+σ[X Y Z]+σ[XY Z]+σ[XY Z ] σ[abc] Let s verify nother exmple: consider the inscribed tringle X 2 Y 2 Z 2 from 23, where X 2 0, 7 4, 11, Y 2 5, 0, 4, Z 2 2, 1, 0 nd 4 X 2 0, 11 4, 7, Y 2 4 4, 0, 5, Z 2 1, 2, 0 The res of the tringles X 2 Y 2Z 2, X 2 Y 2 Z 2 nd X 2 Y 2 Z 2 re So σ[x 2Y Z 2 ] σ[x 2 Y 2Z ] σ[x 2 Y 2 Z 2] , , σ[x 2 Y 2 Z 2 ] + σ[x 2Y 2 Z 2 ] + σ[x 2 Y 2Z 2 ] + σ[x 2 Y 2 Z 2] σ[abc] 2

12 30 Sándor Ngydobi Kiss Theorem 52 If X, Y, Z is the symmetric of X, Y, Z with respect to A, B, C, then σ[x Y Z] σ[xy Z ] σ[cy X] + vw wu, 53 σ[xy Z] σ[x Y Z ] σ[azy ] + wu uv, 54 σ[xy Z ] σ[x Y Z] σ[bxz] + uv vw 55 Proof The res of the tringles AZY, BXZ nd CY X re σ[azy ] w 1 w 0 v1 w v vw, 1 v 0 v σ[bxz] 0 u 1 u w1 u w wu, w 1 w σ[cy X] 1 v 0 v u1 v u uv 0 u 1 u Corollry 51 If X, Y, Z is the symmetric of X, Y, Z with respect to A, B, C then σ[x Y Z] + σ[xy Z] + σ[xy Z ] σ[azy ] + σ[bxz] + σ[cy X], 56 σ[xy Z ] + σ[x Y Z ] + σ[x Y Z] σ[azy ] + σ[bxz] + σ[cy X] 57 References [1] Gómez, BH, The notble configurtion of inscribed equilterl tringles in tringle, Elemente de Mthemtik, , No 1, [2] Equilterl tringle inscribed in tringle, /186432/equilterl-tringle-inscribed-in--tringle [3] Kimberling, C, Encyclopedi of Tringle Centers ETC, edu/ck6/encyclopedi/etchtml CONSTANTIN BRÂNCUŞI TECHNOLOGY LYCEUM SATU MARE, ROMANIA e-mil: dsndorkiss@gmilcom