Neocijenjeno
8. prosinca 2013. 12:12 (10 godine, 7 mjeseci)
Sakrij rješenje
Sakrij rješenje
Zadane su 2 kružnice,
i
, koje se sijeku u točkama
i
. Neka je
pravac određen točkama
i
. Dane su proizvoljne točke
i
s pravca
tako da je dužina
sadržana u dužini
, te tako da vrijedi
. Na kružnici
, odabrane su točke
i
, s iste strane pravca
, takve da su pravci
i
tangente na kružnicu
. Na kružnici
, odabrane su točke
i
, s iste strane pravca
, ali s različite strane u odnosu na točke
i
, takve da su pravci
i
tangente na kružnicu
. Dokažite da se pravci
i
sijeku u točki koja je na pravcu
.
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![\overline{A_1B_1}](/media/m/d/c/2/dc2bd6d9f5e72c0dd287663943db62f3.png)
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
![0 \neq |AA_1| \neq |BB_1| \neq 0](/media/m/0/c/8/0c83df7c2d4943e9fb46cd5984debc5c.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
![BN](/media/m/9/2/3/923313310d49dd4405c2a3573960a679.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
![AR](/media/m/f/3/5/f357f08b4fdbd7fe7522df290756074f.png)
![BS](/media/m/3/0/f/30f1debf47f15e4cdb55f1f18a034c23.png)
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![MN](/media/m/2/6/7/267a73297a5de9e529d41774ee6ff45a.png)
![RS](/media/m/6/5/c/65cfe2e0c3b95a4b9afb102fe26e5ef3.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
Upozorenje: Ovaj zadatak još niste riješili!
Kliknite ovdje kako biste prikazali rješenje.
Kliknite ovdje kako biste prikazali rješenje.
neka je
sjeciste od
i
, te neka je
sjeciste
i
, jasno ova dva pravca nisu paralelna zbog uvjeta
. menelaj na trokut
s obzirom na pravac
nam daje
![\dfrac{|AH|}{|HB|}\dfrac{|BN|}{|NC|}\dfrac{|CM|}{|MA|} = 1](/media/m/4/f/7/4f73cb290172d012d766ddf1cd298d71.png)
kako je ocito
, slijedi
![\dfrac{|AH|}{|HB|} = \dfrac{|MA|}{|BN|}](/media/m/6/5/c/65c01eb78e54e1414022ee246a931859.png)
ista prica sa druge strane, uz
sjeciste
i
daje
![\dfrac{|AG|}{|GB|} = \dfrac{|RA|}{|BS|}](/media/m/0/5/7/05742e8c87f4aff39960bc7dacac913d.png)
ali potencija tocke nam daje
, te
, sto nam daje
![\dfrac{|AH|}{|HB|} = \dfrac{|AG|}{|GB|}](/media/m/4/0/3/403b7b17d03bb0c4429bc929eb2a26af.png)
sto znaci da
i
dijele
u istom omjeru, a buduci da su te dvije tocke s istih strana tocaka i
i
, zakljucujemo da su one medusobno jednake, iz cega slijedi tvrdnja zadatka.
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
![BN](/media/m/9/2/3/923313310d49dd4405c2a3573960a679.png)
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
![MN](/media/m/2/6/7/267a73297a5de9e529d41774ee6ff45a.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![|AA_1| \neq |BB_1|](/media/m/2/1/6/2161b736e7f7d59300d1835512a7aced.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![MN](/media/m/2/6/7/267a73297a5de9e529d41774ee6ff45a.png)
![\dfrac{|AH|}{|HB|}\dfrac{|BN|}{|NC|}\dfrac{|CM|}{|MA|} = 1](/media/m/4/f/7/4f73cb290172d012d766ddf1cd298d71.png)
kako je ocito
![|CN| = |CM|](/media/m/6/f/a/6fa6a8b4d06fe2a60af54b31fdb2e2a5.png)
![\dfrac{|AH|}{|HB|} = \dfrac{|MA|}{|BN|}](/media/m/6/5/c/65c01eb78e54e1414022ee246a931859.png)
ista prica sa druge strane, uz
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![RS](/media/m/6/5/c/65cfe2e0c3b95a4b9afb102fe26e5ef3.png)
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
![\dfrac{|AG|}{|GB|} = \dfrac{|RA|}{|BS|}](/media/m/0/5/7/05742e8c87f4aff39960bc7dacac913d.png)
ali potencija tocke nam daje
![|AM|^2 = |AA_1||AB_1| = |AR|^2](/media/m/c/6/8/c688f5bcd9694cb30fc304e277a9eadd.png)
![|BN|^2 = |BB_1||BA_1| = |BS|^2](/media/m/a/5/8/a58f701316949a2e5a2e8232511880b1.png)
![\dfrac{|AH|}{|HB|} = \dfrac{|AG|}{|GB|}](/media/m/4/0/3/403b7b17d03bb0c4429bc929eb2a26af.png)
sto znaci da
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)