1. (2 п) Доказати да се тежиште тетраедра
![ABCD ABCD](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/e1ea6a815485d05f1c11d65a89209fae.png)
поклапа са тежиштем тетраедра
![A'B'C'D' A'B'C'D'](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/3dc1369491ba8b6bcaf9c5aca5dd8e96.png)
коме су темена
![A',B',C',D' A',B',C',D'](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/a9264006097fd6402417b7842078254f.png)
редом тежишта троуглова
![BCD BCD](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/5b87b1178a998873bba9519c42eb9157.png)
,
![ACD ACD](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/fcb3bb8320953285e8933d5769ac808a.png)
,
![ABD ABD](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/7f5bc82ee34b66e01b561ab650f589e7.png)
и
![ABC ABC](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/c5128f579b83322a464b5b5065364dd8.png)
.
2. (2 п) Нека је
![ABC ABC](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/c5128f579b83322a464b5b5065364dd8.png)
троугао и тачке
![P P](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/418ddcc4ac5d1150e5b90cce6b71e14c.png)
и
![Q Q](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/3ec7122626586a9a4462755e4b66ea40.png)
такве да је
![\overrightarrow{AP}=\frac{1}{2}\overrightarrow{PB} \overrightarrow{AP}=\frac{1}{2}\overrightarrow{PB}](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/7a68751af744ce51c3cb02992bcb388a.png)
и
![\overrightarrow{BQ} = 4\overrightarrow{QC} \overrightarrow{BQ} = 4\overrightarrow{QC}](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/5f48b9d7d14c3cc33c327b418d3a0324.png)
, а тачка
![R R](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/75695b46abca7ce53dfa3b4e984a45ca.png)
је пресек правих
![AC AC](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/35ed99217933dfe09b0194e1a3251ec2.png)
и
![PQ PQ](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/a2a0a4e85fa85b2e8751982d18207526.png)
. Израчунати однос
![\overrightarrow{CR}:\overrightarrow{RA} \overrightarrow{CR}:\overrightarrow{RA}](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/2dbb4ef62cea0467a54b4340c2601988.png)
.
3. (2 п) У равни је дат троугао
![ABC ABC](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/c5128f579b83322a464b5b5065364dd8.png)
. Нека тачка
![D D](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/2f4396bab5869c1e0c9f8a7620bf2518.png)
припада страници
![AB AB](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/adfda2c63a6b4f70f610eb963324646d.png)
, а тачка
![E E](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/6228f7994a21ee53499c6684fac51774.png)
страници
![BC BC](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/886a2f4b696ad3881ed755fd2f5ea71b.png)
тако да је
![\frac{AD}{DB}=\frac{3}{4} \frac{AD}{DB}=\frac{3}{4}](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/2bbe57620189d28bf354cfe7f23253c3.png)
и
![\frac{BE}{EC}=\frac{5}{7} \frac{BE}{EC}=\frac{5}{7}](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/a642d9a91f5e0cb2b41d574ecffbaa44.png)
. Ако се дужи
![AE AE](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/2dcd46962d6f33a77086cc4e7460b33e.png)
и
![CD CD](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/e77dead140baa684b3f569793270918a.png)
секу у тачки F одредити у ком односу тачка
![F F](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/72af65f44f723b95d86b5f63f7c3ee77.png)
дели дужи
![AE AE](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/2dcd46962d6f33a77086cc4e7460b33e.png)
и
![CD CD](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/e77dead140baa684b3f569793270918a.png)
.
4. (2 п) Нека је
![ABC ABC](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/c5128f579b83322a464b5b5065364dd8.png)
троугао,
![P P](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/418ddcc4ac5d1150e5b90cce6b71e14c.png)
и
![Q Q](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/3ec7122626586a9a4462755e4b66ea40.png)
тачке такве да је
![3\overrightarrow{AP} = \overrightarrow{BA} 3\overrightarrow{AP} = \overrightarrow{BA}](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/df1d0f456e7bad667b5f5a8e7388f8b0.png)
и
![2\overrightarrow{BQ}=\overrightarrow{BC} 2\overrightarrow{BQ}=\overrightarrow{BC}](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/8875d58f0e72d28b9b1bbf850b911f5f.png)
. Ако је
![R R](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/75695b46abca7ce53dfa3b4e984a45ca.png)
тачка праве
![AC AC](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/35ed99217933dfe09b0194e1a3251ec2.png)
таква да се праве
![AQ AQ](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/14428e91495f1dd474c0942b3c9fa005.png)
,
![CP CP](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/95364f3d158439e0f94477c231a9c29d.png)
и
![BR BR](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/7ec825e04bb05f1e49b605684b827614.png)
секу у једној тачки, одредити однос
![\overrightarrow{AC}:\overrightarrow{AR} \overrightarrow{AC}:\overrightarrow{AR}](https://elearning.rcub.bg.ac.rs/moodle/filter/tex/pix.php/bc6487b09684e0de891d4417132f7e28.png)
.