<prompt> Katika pembetatu ABC, a 7, b 9, na c 4. Acha I iwe kitovu.asyukubwa wa kitengo sentimita 0.8;jozi A, B, C, D, E, F, I;B 0,0;C 7,0;A sehemu ya makutanoarcB,4,0,180,arcC,9,0,180;I kitovuA,B,C;kuchoraA--B--C--mzunguko;choraduaraA,B,C;leboA, A, N;leboB, B, SW;leboC, C, SE;nukta I, I, NE;asyKishaoverrightarrowI x overrightarrowA y overrightarrowB z overrightarrowC,ambapo x, y, na z ni viambajengo hivi kwamba x y z 1. Ingiza x,y,z iliyoagizwa mara tatu.  <response>: Tunajua kuwa I iko kwenye viambatanisho vya pembe overlineAD, overlineBE, na overlineCF.asyukubwa wa kitengo sentimita 0.8;jozi A, B, C, D, E, F, I;B 0,0;C 7,0;A sehemu ya makutanoarcB,4,0,180,arcC,9,0,180;I kitovuA,B,C;D uganiA, I, B, C;E uganiB, I, C, A;F uganiC, I, A, B;kuchoraA--B--C--mzunguko;kuchoraA--D;kuchoraB--E;kuchoraC--F;leboA, A, N;leboB, B, SW;leboC, C, S;leboD, D, S;leboE, E, NE;leboF, F, SW;leboI, I, S;asyKwa Nadharia ya Angle Bisector, BD:DC AB:AC 4:9, hivyooverrightarrowD frac913 overrightarrowB frac413 overrightarrowC.Pia, na Angle Bisector Theorem, CE:EA BC:AB 7:4, hivyooverrightarrowE frac411 overrightarrowC frac711 overrightarrowA.Kutenga overrightarrowC katika kila mlinganyo, tunapataoverrightarrowC frac13 overrightarrowD - 9 overrightarrowB4 frac11 overrightarrowE - 7 overrightarrowA4. Kisha 13 overrightarrowD - 9 overrightarrowB 11 overrightarrowE - 7 overrightarrowA, au 13 overrightarrowD 7 overrightarrowA 11 overrightarrow E 9 mshale wa kuliaB, aufrac1320 overrightarrowD frac720 overrightarrowA frac1120 overrightarrowE frac9101 20 overrightarrowB.Kwa kuwa vihesabu vya pande zote mbili za mlingano vinaongeza hadi 1, vekta iliyo upande wa kushoto iko kwenye mstari AD, na vekta iliyo upande wa kulia iko kwenye mstari BE. Kwa hivyo, vekta hii ya kawaida ni overrightarrowI. KishaanzaalignoverrightarrowI frac1320 overrightarrowD frac720 overrightarrowA frac1320 left frac913 overrightarrowB frac413 overrightarrowC right frac7101 20 mshale wa kulia zaidiA frac720 overrightarrowA frac920 overrightarrowB frac15 overrightarrowC.endalignHivyo, x,y,z left frac720, frac920, frac15 kulia. Kwa ujumla zaidi, kitovu I cha pembetatu ABC hutosheleza kila wakatioverrightarrowI fracaa b c overrightarrowA fracba b c overrightarrowB fracc a b c mshale wa kuliaC. 