Kulolu shicilelo, sizocubungula enye yemibono esemqoka ye-Euclidean geometry – theorem kaStewart, eyathola leli gama ngokuhlonipha isazi sezibalo saseNgilandi uM. Stewart, owakufakazela lokho. Sizophinde sihlaziye ngokuningiliziwe isibonelo sokuxazulula inkinga ukuze sihlanganise indaba ethulwe.
Isitatimende sethiyori
Unxantathu kaDan ABC. Eceleni kwakhe AC iphuzu lithathiwe D, exhunywe phezulu B. Samukela isaziso esilandelayo:
- AB = a
- BC = b
- BD = p
- AD = x
- DC = kanye
Kulonxantathu, ukulingana kuyiqiniso:
Ukusetshenziswa kwe-theorem
Kusukela ku-theorem ka-Stewart, amafomula angatholakala ukuze uthole ama-medians nama-bisectors kanxantathu:
1. Ubude be-bisector
Ake lc isinqamu esiphindwe kabili sidonseleka eceleni c, ehlukaniswe yaba izingxenye x и y. Ake sithathe ezinye izinhlangothi ezimbili zikanxantathu njenge a и b… Esimweni esinjalo:
2. Ubude obumaphakathi
Ake mc i-median yehliselwe eceleni c. Ake sisho ezinye izinhlangothi ezimbili zikanxantathu ngokuthi a и b… Bese:
Isibonelo senkinga
Unxantathu unikiwe I-ABC. Ehlangothini I-AC ilingana no-9 cm, iphuzu lithathiwe D, ehlukanisa uhlangothi ukuze AD kabili ubude DC. Ubude bengxenye exhuma i-vertex B futhi iphuzu D,u5cm. Kulokhu, unxantathu owenziwe ABD i-isosceles. Thola izinhlangothi ezisele zikanxantathu ABC.
Isixazululo
Ake sibonise izimo zenkinga ngendlela yomdwebo.
AC = AD + DC = 9cm. AD eside DC kabili, isib AD = I-2DC.
Ngenxa yalokho, i- I-2DC + DC = I-3DC u9d XNUMX cm. Ngakho, DC = 3cm, AD = 6cm.
Ngoba unxantathu ABD - isosceles, nohlangothi AD 6 cm, ngakho ayalingana AB и BDIe AB = 5cm.
Kusele ukuthola kuphela BC, ithola ifomula ku-theorem kaStewart:
Sifaka esikhundleni samanani aziwayo kule nkulumo:
Ngale ndlela, BC = √52 ≈ 7,21 cm.