Ceva's theorem: formulation and example with solution

Hauv kev tshaj tawm no, peb yuav xav txog ib qho ntawm cov theorems classical ntawm affine geometry - Ceva theorem, uas tau txais lub npe zoo li no los ntawm Italian engineer Giovanni Ceva. Peb kuj tseem yuav tshuaj xyuas ib qho piv txwv ntawm kev daws qhov teeb meem txhawm rau txhawm rau sib sau cov ntaub ntawv nthuav tawm.

Cov ntsiab lus

Nqe lus ntawm lub theorem

Daim duab peb sab muab ABC, nyob rau hauv uas txhua vertex yog txuas mus rau ib tug taw tes ntawm opposite sab.

Cevas theorem: formulation and example with solution

Yog li, peb tau txais peb ntu (AA', BB ' и CC '), uas hu ua cev.

Cov ntu no cuam tshuam ntawm ib qho taw tes yog tias thiab tsuas yog tias muaj kev sib npaug hauv qab no:

|THIAB'| |TSIS'| |CB'| = |BC'| |SHIFT'| |AB'|

Lub theorem tseem tuaj yeem nthuav tawm hauv daim ntawv no (nws tau txiav txim siab nyob rau hauv qhov piv ntawm cov ntsiab lus faib rau sab):

Cevas theorem: formulation and example with solution

Ceva's trigonometric theorem

Cevas theorem: formulation and example with solution

Nco tseg: tag nrho cov ces kaum yog oriented.

Piv txwv ntawm qhov teeb meem

Daim duab peb sab muab ABC nrog dots TO', B' и VS ' ntawm ob sab BC, AC и AB, raws. Lub vertices ntawm daim duab peb sab txuas nrog rau cov ntsiab lus muab, thiab cov ntu tsim dhau los ntawm ib qho taw tes. Tib lub sijhawm, cov ntsiab lus TO' и B' coj nyob rau hauv lub midpoints ntawm tus coj opposite sab. Nrhiav seb qhov piv ntawm qhov point VS ' faib sab AB.

tshuaj

Cia peb kos ib daim duab raws li cov xwm txheej ntawm qhov teeb meem. Rau peb txoj kev yooj yim, peb txais yuav cov lus hauv qab no:

  • AB = B'C = a
  • AB = A'C = b

Cevas theorem: formulation and example with solution

Nws tseem tsuas yog tsim qhov sib piv ntawm ntu raws li Ceva theorem thiab hloov cov ntawv lees paub rau hauv nws:

Cevas theorem: formulation and example with solution

Tom qab txo cov feem, peb tau txais:

Cevas theorem: formulation and example with solution

Yog li, AC = AB,ie point VS ' faib sab AB hauv ib nrab.

Yog li ntawd, hauv peb daim duab peb sab, cov ntu AA', BB ' и CC ' yog medians. Tau daws qhov teeb meem, peb tau ua pov thawj tias lawv sib tshuam ntawm ib qho chaw ( siv tau rau txhua daim duab peb sab).

Nco ntsoov: siv Ceva's theorem, ib tus tuaj yeem ua pov thawj tias hauv daim duab peb sab ntawm ib qho taw tes, cov bisectors lossis qhov siab kuj sib tshuam.

Sau ntawv cia Ncua