U konveksnom četverokutu
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
točke
![G_1](/media/m/b/c/d/bcd9cf8ee60f35cdf346a1ce12a31bc1.png)
,
![G_2](/media/m/8/a/6/8a68fbfad95710d3eddbccfae32746f0.png)
,
![G_3](/media/m/d/2/c/d2c097823396c63586832981dec168a3.png)
,
![G_4](/media/m/2/d/6/2d6677e09fedefc97ccd24b80ca7bde6.png)
su redom težišta trokuta
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
,
![ACD](/media/m/0/b/1/0b171034d79122bd02f64bc8f6ae94dd.png)
,
![ABD](/media/m/a/5/4/a548bc577543629d304ecba1a042f910.png)
,
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, dok su
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
,
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
,
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
,
![D_1](/media/m/f/e/6/fe67388584f844e56a8db45e4e8768ca.png)
, točke centralno simetrične točkama
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
,
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
u odnosu na
![G_1](/media/m/b/c/d/bcd9cf8ee60f35cdf346a1ce12a31bc1.png)
,
![G_2](/media/m/8/a/6/8a68fbfad95710d3eddbccfae32746f0.png)
,
![G_3](/media/m/d/2/c/d2c097823396c63586832981dec168a3.png)
,
![G_4](/media/m/2/d/6/2d6677e09fedefc97ccd24b80ca7bde6.png)
. Dokažite da je
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
paralelogram ako i samo ako je
![A_1B_1C_1D_1](/media/m/8/e/a/8ea8991888072519f65b0a7e2f45de2d.png)
paralelogram.
%V0
U konveksnom četverokutu $ABCD$ točke $G_1$, $G_2$, $G_3$, $G_4$ su redom težišta trokuta $BCD$, $ACD$, $ABD$, $ABC$, dok su $A_1$, $B_1$, $C_1$, $D_1$, točke centralno simetrične točkama $A$, $B$, $C$, $D$ u odnosu na $G_1$, $G_2$, $G_3$, $G_4$. Dokažite da je $ABCD$ paralelogram ako i samo ako je $A_1B_1C_1D_1$ paralelogram.