Stewart's theorem: kuumbwa uye muenzaniso nemhinduro

Muchinyorwa chino, tichatarisa imwe yedzidziso huru dzeEuclidean geometry - Stewart's theorem, iyo yakagamuchira zita rakadaro mukuremekedza nyanzvi yemasvomhu yechiRungu M. Stewart, akazvipupurira. Isu tichaongororawo zvakadzama muenzaniso wekugadzirisa dambudziko rekubatanidza izvo zvakaratidzwa.

Chirevo che theorem

Dan Triangle ABC. Padivi pake AC pfungwa yakatorwa D, iyo yakabatana nepamusoro B. Tinobvuma chirevo chinotevera:

• AB = a
• BC = b
• BD = p
• AD = x
• DC = uye

Kune iyi katatu, kuenzana ichokwadi:

Kushandiswa kwe theorem

Kubva kuStewart's theorem, mafomula anogona kutorwa ekutsvaga ma medians uye bisectors etatu:

1. Hurefu hwebhizi

Regai l_c ndiye bisector yakadhonzwa kudivi c, iyo yakakamurwa kuita zvikamu x и y. Ngatitore mamwe mativi maviri egonyo se a и b… Muchiitiko ichi:

2. Kureba kweMedhia

Regai m_c ndiye wepakati wakadzikiswa parutivi c. Ngatiratidze mamwe mativi maviri egonyo se a и b… Zvadaro:

Muenzaniso wedambudziko

Triangle yapiwa ABC. Padivi AC yakaenzana ne9cm, pfungwa yakatorwa D, iyo inokamura rutivi kuti AD zvakapetwa kaviri DC. Hurefu hwechikamu chinobatanidza vertex B uye point D, i5cm. Muchiitiko ichi, iyo katatu yakaumbwa US iri isosceles. Tsvaga mativi asara egonyonhatu ABC.

mhinduro

Ngatiratidze mamiriro edambudziko nenzira yekudhirowa.

AC = AD + DC = 9cm. AD kwenguva DC kaviri, kureva AD = 2DC.

Nekudaro, iyo 2DC + DC = 3DC u9d XNUMX cm. Saka, DC = 3cm, AD = 6cm.

Nokuti katatu US - isosceles, uye rutivi AD ari 6 cm, saka akaenzana AB и BDIe AB = 5cm.

Inoramba chete kuwana BC, kutora fomula kubva kuStewart's theorem:

Isu tinotsiva izvo zvinozivikanwa mukutaura uku:

Nenzira iyi, BC = √‎52 ≈ 7,21 cm.