Kamp '13 - Geometrija 12.
Dodao/la:
arhiva3. studenoga 2013. Neka je
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
trokut i
![J](/media/m/9/0/e/90ef5cc2558381e341da5808eb92126f.png)
središte tom trokutu pripisane kružnice nasuprot vrhu
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
. Ta pripisana kružnica dodiruje
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
u
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
, a pravce
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
i
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
redom u
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
i
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
. Pravci
![LM](/media/m/c/4/a/c4aa56ea011437258c94dd4925ad7c32.png)
i
![BJ](/media/m/2/0/d/20d7c940a45a6c54004a12ffc95b40ef.png)
sijeku se u
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
, a
![KM](/media/m/0/1/c/01ca9badbb81b31fdf7b3e19f0f0c6c6.png)
i
![CJ](/media/m/d/6/a/d6a20690e5234f6fa2d642edb68b2505.png)
u
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
. Neka je
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
sjecište
![AF](/media/m/a/e/4/ae455e708e936870cb86e6a074a2c5a0.png)
i
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
i neka je
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
sjecište
![AG](/media/m/7/2/4/724f04bf00575e4848fa5c7dc9cb26af.png)
i
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. Dokaži da je
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
polovište
![ST](/media/m/f/7/4/f74405e2fa689cb992ff2ea89c8cf0e3.png)
.
%V0
Neka je $ABC$ trokut i $J$ središte tom trokutu pripisane kružnice nasuprot vrhu $A$. Ta pripisana kružnica dodiruje $BC$ u $M$, a pravce $AB$ i $AC$ redom u $K$ i $L$. Pravci $LM$ i $BJ$ sijeku se u $F$, a $KM$ i $CJ$ u $G$. Neka je $S$ sjecište $AF$ i $BC$ i neka je $T$ sjecište $AG$ i $BC$. Dokaži da je $M$ polovište $ST$.
Izvor: Kamp 2013. - Geometrija, M. M.
Komentari:
lkkraljevic, 8. ožujka 2018. 09:14
Shortlist
Shortlist $2012- G1$