<prompt> Katika pembetatu ABC, a 8, b 7, na c 5. Acha H iwe kituo cha orthocenter.asyukubwa wa kitengo cm 0.6;jozi A, B, C, D, E, F, H;B 0,0;C 8,0;A sehemu ya makutanoarcB,5,0,180,arcC,7,0,180;H kituo cha orthocenterA,B,C;D A tafakariB,CA2;E B tafakariC,AB2;F C tafakariA,BC2;kuchoraA--B--C--mzunguko;kuchoraA--D,dashed;kuchoraB--E,dashed;kuchoraC--F,dashed;leboA, A, N;leboB, B, SW;leboC, C, SE;leboH, H, SE, UnFill;nukta H;asyKishaoverrightarrowH x overrightarrowA y overrightarrowB z overrightarrowC,ambapo x, y, na z ni vitu visivyobadilika hivi kwamba x y z 1. Ingiza x,y,z iliyoagizwa mara tatu.  <response>: Acha miinuko iwe overlineAD, overlineBE, na overlineCF.asyukubwa wa kitengo cm 0.6;jozi A, B, C, D, E, F, H;B 0,0;C 8,0;A sehemu ya makutanoarcB,5,0,180,arcC,7,0,180;H kituo cha orthocenterA,B,C;D A tafakariB,CA2;E B tafakariC,AB2;F C tafakariA,BC2;kuchoraA--B--C--mzunguko;kuchoraA--D,dashed;kuchoraB--E,dashed;kuchoraC--F,dashed;leboA, A, N;leboB, B, SW;leboC, C, SE;leboD, D, S;leboE, E, NE;leboF, F, NW;leboH, H, SE, UnFill;nukta H;asyKwa Sheria ya Cosines,anzaaligncos A frac52 72 - 822 cdot 5 cdot 7 frac17, cos B frac52 82 - 722 cdot 5 cdot 8 frac12, cos C frac72 82 - 522 cdot 7 cdot 8 frac1114.endalignKisha BD AB cos B frac52 na CD AC cos C frac112, hivyooverrightarrowD frac1116 overrightarrowB frac516 overrightarrowC.Pia, AE AB cos A frac 57 na CE BC cos C frac447, hivyooverrightarrowE frac4449 overrightarrowA frac549 overrightarrowC.Kutenga overrightarrowC katika milinganyo hii, tunapataoverrightarrowC frac16 overrightarrowD - 11 overrightarrowB5 frac49 overrightarrowE - 44 overrightarrowA5. Kisha 16 overrightarrowD - 11 overrightarrowB 49 overrightarrowE - 44 overrightarrowA, hivyo 16 overrightarrowD 44 overrightarrowA 49 overrightarrow E 11 mshale wa kuliaB, aufrac1660 overrightarrowD frac4460 overrightarrowA frac4960 overrightarrowE frac11101 60 mshale uliopitilizaB.Kwa vile viambatanishi 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 overrightarrowH. KishaanzaalignoverrightarrowH frac4960 overrightarrowE frac1160 overrightarrowB frac4960 left frac4449 overrightarrowA frac549 overrightarrowC kulia frac11101 60 mshale wa kuliaB frac1115 overrightarrowA frac1160 overrightarrowB frac112 overrightarrowC.endalignHivyo, x,y,z left frac1115, frac1160, frac112 kulia. 