Nad stranicama trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
konstruirani su slični trokuti
![ABD](/media/m/a/5/4/a548bc577543629d304ecba1a042f910.png)
,
![BCE](/media/m/b/d/0/bd0a872febc285929044d2c5126a6005.png)
,
![CAF](/media/m/1/c/0/1c095cefdead960d4773e7ea80f1b446.png)
(
![k = |AD|:|DB| = |BE|:|EC| = |CF|:|FA|](/media/m/b/d/a/bdac2fd426bfc214f2cb9b370cd4d505.png)
;
![\alpha = \angle ADB = \angle BEC = \angle CFA](/media/m/5/9/5/59547a8142b26aad091567316fb3e864.png)
). Dokažite da su polovišta dužina
![\overline{AC}](/media/m/d/9/5/d95354f0f833a5fda9c16a01a878c14f.png)
,
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
,
![\overline{CD}](/media/m/3/3/8/338870e40f3ea7992d83158230115a5f.png)
i
![\overline{EF}](/media/m/7/3/6/736526ec2c1c20572842175dc3523f2c.png)
vrhovi paralelograma, čiji je jedan kut jednak
![\alpha](/media/m/f/c/3/fc35d340e96ae7906bf381cae06e4d59.png)
, a omjer duljina odgovarajućih stranica
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
.
%V0
Nad stranicama trokuta $ABC$ konstruirani su slični trokuti $ABD$, $BCE$, $CAF$ ($k = |AD|:|DB| = |BE|:|EC| = |CF|:|FA|$; $\alpha = \angle ADB = \angle BEC = \angle CFA$). Dokažite da su polovišta dužina $\overline{AC}$, $\overline{BC}$, $\overline{CD}$ i $\overline{EF}$ vrhovi paralelograma, čiji je jedan kut jednak $\alpha$, a omjer duljina odgovarajućih stranica $k$.