Forum Geometriorum Volume 4 004 15 134. FORUM GEOM ISSN 1534-1178 The Intouh Tringle nd the OI-line Eri Dnneel Abtrt. We prove ome intereting reult relting the intouh tringle nd the OI line of tringle. We lo give ome intereting propertie of the tringle enter X 57 the homotheti enter of the intouh nd exentrl tringle. 1. Introdution L. Emelynov [4] h reently given n intereting reltion between the OI-line nd the tringle of refletion of the intouh tringle. Here O nd I re repetively the irumenter nd inenter of the tringle. Given tringle ABC with intouh tringle XY Z let X Y Z be the refletion of X Y Z in their repetive oppoite ide YZ ZX XY. Then the line AX BY CZ interet BC CA AB t the interept of the OI-line. A X Y Z X 7 I X 144 O Z B X C Y Figure 1. Emelynov [3] lo noted tht the interept of the point IX BC IY CA IZ AB form tringle perpetive with ABC. See Figure 1. Aording to [7] thi perpetor i the point b + b + X 144 = : b + + b Publition Dte: Augut 10 004. Communiting Editor: Pul Yiu. The uthor thnk Pul Yiu for hi help in the preprtion of thi pper. : + b + b
16 E. Dnneel on the Soddy line joining the inenter nd the Gergonne point. In thi pper we generlize thee reult. We work with bryentri oordinte with referene to tringle ABC.. The tringle enter X 57 Let b be the length of the ide BC CA AB of tringle ABC nd = 1 + b + the emiperimeter. The intouh tringle XY Z nd the exentrl tringle with the exenter vertie re lerly homotheti ine their orreponding ide re perpendiulr to the me ngle bietor of tringle ABC. Thee tringle re repetively the evin tringle of the Gergonne point 1 : 1 b : 1 nd the ntievin tringle of the inenter : b : their homotheti enter h oordinte +b+ : : = : : = : :. Thi i the tringle enter X 57 in [6] defined the iogonl onjugte of the Mittenpunkt X 9 = :b :. Thi i point on the OI-line ine the two tringle in quetion hve irumenter I nd X 40 the refletion of I in O 1 We give ome intereting propertie of the tringle X 57. Sine ABC i the orthi tringle of the exentrl tringle it i homotheti to the orthi tringle X 1 Y 1 Z 1 of XY Z with the me homotheti enter X 57. See Figure. A Y X 1 Z X 57 I Y 1 Z 1 O B X C Figure. 1 The irumirle of ABC i the nine-point irle of the exentrl tringle.
The intouh tringle nd the OI-line 17 Let DEF be the irumevin tringle of the inenter I nd D E F the ntipode of D E F in the irumirle. In other word D nd D re the midpoint of the two r BC D on the r ontining the vertex A; imilrly for the other two pir. Clerly D = b + : b b : = : bb + :b +. Similrly E = + : b : + nd F = + b :b + b :. To ompute the oordinte of D E F we mke ue of the following formul. Lemm 1. Let P = vw : b wu : uv be point on the irumirle o tht u + v + w =0. For point Q =x : y : z different from P nd not lying on the irumirle the line PQ interet the irumirle gin t the point vw + tx : b wu + ty : uv + tz where t = b u x + v y + b w z yz + b zx +. 1 xy Proof. Entering the oordinte X Y Z = vw + tx : b wu + ty : uv + tz into the eqution of the irumirle YZ + b ZX + XY =0 we obtin yz + b zx + xyt +b uv + wx + vw + uy + b wu + vzt + b uvwu + v + w =0. Sine u + v + w =0 thi give t =0or the vlue given in 1 bove. Let M =0:1:1be the midpoint of BC. Applying Lemm 1 to D nd M we obtin D = : bb : b. Similrly E = : b : nd F = b :bb :. Applying Lemm 1 to D nd X =0: + b : + b likewie to E nd Y nd to F nd Z we obtin the point X = b + b : b + b : + b Y = b + : b b + : + b Z = b + : b + b : + b b.
18 E. Dnneel Thee re lerly the vertie of the irumevin tringle of X 57. We ummrize thi in the following propoition. Propoition. If X repetively Y Z re the eond interetion of D X repetively E Y F Z nd the irumirle then X Y Z i the irumevin tringle of X 57. D A Y E F Y Z Z X 57 I O F B E X C X D Figure 3. Remrk. The line D X E Y F Z interet t X 55 the internl enter of imilitude of the irumirle nd the inirle. Propoition 3. Let X Y Z be the eond interetion of the irumirle with the line DX EY FZ repetively. The line AX BY CZ bound the ntievin tringle of X 57. Proof. By Lemm 1 thee re the point X bb = : : Y = : b b Z = : bb The line AX BY CZ hve eqution b : :. b b y + z = 0 x + z = 0 x + b b y = 0. They lerly bound the ntievin tringle of X 57. See Figure 4.
The intouh tringle nd the OI-line 19 X A E F Y Y B Z X 57 I X O C Z D Figure 4. Remrk. The line DX EY FZ interet t X 56 the externl enter of imilitude of the irumirle nd inirle. Propoition 4. X 57 i the perpetor of the tringle bounded by the polr of A B C with repet to the irle through the exenter. Proof. A i eily verified the eqution of the irumirle of the exentrl tringle i yz + b zx + xy +x + y + zbx + y + bz =0. The polr re the line They bound tringle with vertie x + y b + z = 0 x + y + z = 0 x + y b + z = 0.
130 E. Dnneel Thi lerly h perpetor X 57. b : b : : b : : : b Propoition 5. X 57 i the perpetor of the refletion of the Gergonne point in the intouh tringle.. A Y Z X 57 O B X C Figure 5. More generlly the refletion tringle of P =u : v : w in the evin tringle of P i perpetive with ABC t u u + b v + w + b + : :. vw See []. For exmple if P i the inenter thi perpetor i the point X 35 = b + b + :b + + b : + b + b whih divide the egment OI in the rtio OX 35 : X 35 I = R :r. Finlly we lo mention from [5] tht X 57 i the orthoorrepondent of the inenter. Thi men tht the triliner polr of X 57 nmely the line x + b y + z =0
The intouh tringle nd the OI-line 131 interet the ideline BC CA AB t X Y Z repetively uh tht IX IA IY IB nd IZ IC. 3. A lou of perpetor A n extenion of the reult of [4] we onider for rel number t the tringle X t Y t Z t with X t Y t Z t dividing the egment XX 1 YY 1 ZZ 1 in the rtio XX t : X t X 1 = YY t : Y t Y 1 = ZZ t : Z t Z 1 = t :1 t. Propoition 6. The tringle X t Y t Z t i perpetive with ABC. The lou of the perpetor i the Soddy line joining the inenter to the Gergonne point. Proof. We ompute the oordinte of X t Y t Z t. It i well known tht BX = XC = et. o tht in bolute bryentri oordinte B + bc C + A A + B X = Y = Z =. b Sine the intouh tringle XY Z h ute ngle B+C C+A nd A+B t X Y Z repetively the pedl X 1 of X on YZdivide the egment in the rtio YX 1 : X 1 Z =ot C + A :ot A + B =tn B :tnc = :. Similrly Y 1 nd Z 1 divide ZX nd XY in the rtio ZY 1 : Y 1 X = : XZ 1 : Z 1 Y = :. In bolute bryentri oordinte Y + Z X 1 = b + A + bb + C =. b It follow tht X t =1 tx + tx 1 tb + A + b tb + b tc = b In homogeneou bryentri oordinte thi i X t =tb + :b t : b t.. The line IX t h eqution bb x + b ty b tz =0. The line IX t interet BC t the point X t =0 : b t :b t b t b t = 0: :.
13 E. Dnneel Similrly the line IY t nd IZ t interet CA nd AB repetively t b t Y t b t = :0: b t Z t = b t : :0. The tringle X t Y i perpetive with ABC t the point t Z t b t b t b t : :. A t vrie thi perpetor trvere tright line. Sine the perpetor i the Gergonne point for t =0nd the inenter for t = thi line i the Soddy line joining thee two point. The Soddy line h eqution b x + y + b z =0. Here re ome tringle enter on the Soddy line with the orreponding vlue of t. The ymbol r tnd for the rdiu of the A-exirle. t perpetor firt bryentri oordinte b 1 X + 77 b X +b+ 144 1 X 69 R X 481 r R X 48 r + R X 175 r R X 176 r + 3R X 137 4r 3 3R X 1371 4r +3 R X 1374 4r R X 1373 4r + The infinite point of the Soddy point i the point X 516 = 3 b+ +b :b 3 +b + : 3 +b + b. It orrepond to t = R4R+r. The delonghmp point X 0 lo lie on the Soddy line. It orrepond to t = RR+r. 4. Emelynov firt problem From the oordinte of X t we eily find the interetion A t = AX t BC B t = BX t CA C t = CX t AB.
The intouh tringle nd the OI-line 133 Thee re A t =0 : b t :b t B t = t :0: t C t =b t :b t :0. They re olliner if nd only if tb t t + tb t t =0. 3 Sine thi i ubi eqution in t there re three vlue of t for whih A t B t C t re olliner. One of thee i t =ording to [4]. The other two root re given by b bt +t =0. 4 Sine b =4Rr nd b =r where R nd r re repetively the irumrdiu nd inrdiu thi beome From thi R Rt + rt =0. 5 t = R ± R Rr r = R ± d r where d i the ditne between O nd I. We identify the line orreponding to thee two vlue of t. Propoition 7. Correponding to the two root of 4 the line ontining A t B t C t re the tngent to the inirle perpendiulr to the OI-line. Lemm 8. Conider tringle ABC with intouh tringle XY Z nd line L intereting the ide BC CA AB t A B C repetively. The line L i tngent to the inirle if nd only if one of the following ondition hold. 1 The interetion BB CC lie on the line YZ. The interetion CC AA lie on the line ZX. 3 The interetion AA BB lie on the line XY. Proof. Let A B be tngent to the inirle. By Brinhon theorem pplied to the irumribed hexgon AY B A XB it immeditely follow tht AA YXnd B B re onurrent. Now uppoe AA YXnd B B re onurrent. Conider the tngent through A different from BC to the inirle. Let B be the interetion of thi tngent with AC. It follow from the preeeding tht AA YX nd B B re onurrent. Therefore B mut oinide with B. Thi men tht A B i tngent to the inirle.
134 E. Dnneel 5. Proof of Propoition 7 The line BB t nd CC t interet t the point A = b t t b : t t : tb t. Thi point lie on the line YZ : x + by + z =0if nd only if b t t + b t t + tb t =0. Thi redue to eqution 4 bove. By Lemm 8 thee two line re tngent to the inirle. We lim tht thee re the tngent perpendiulr to the line OI. From the oordinte given in the eqution of the line B t C t i t t x tb t + y b t t + z =0. Aording to [6] line perpendiulr to OI hve infinite point X 513 =b :b : b. The line B t C t ontin the infinite point X 513 if nd only if the me eqution 4 hold. Thi how tht the two line in quetion re indeed the tngent to the inirle perpendiulr to the OI-line. Referene [1] E. Dnneel Hyintho mege 8670 November 18 003. [] J.-P. Ehrmnn Hyintho mege 7999 September 4 003. [3] L. Emelynov Hyintho mege 8645 November 16 003. [4] L. Emelynov On the interept of the OI-line Forum Geom. 4 004 81 84. [5] B. Gibert Orthoorrepondene nd orthopivotl ubi Forum Geom. 3 003 1 7. [6] C. Kimberling Enylopedi of Tringle Center vilble t http://fulty.evnville.edu/k6/enylopedi/etc.html. [7] P. Yiu Hyintho mege 8647 November 16 003. [8] P. Yiu Hyintho mege 8675 November 18 003. Eri Dnneel: Hubert d Ydewlletrt 6 8730 Beernem Belgium E-mil ddre: eri.dnneel@pndor.be