Neka je točka
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
središte opisane kružnice trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
s kutovima
![\alpha=\angle{BAC}](/media/m/d/3/3/d337edb855490f27428baf5b1dbd848b.png)
i
![\beta = \angle{CBA}](/media/m/3/1/b/31b364174f2ea32f3e1ccc5b9f10a347.png)
. Neka pravac
![CS](/media/m/f/6/f/f6fa7069052ebcf50b7ffc5edb90eabf.png)
siječe pravac
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
u točki
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
koja se nalazi između točaka
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
i
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
. Dokaži da vrijedi
%V0
Neka je točka $S$ središte opisane kružnice trokuta $ABC$ s kutovima $\alpha=\angle{BAC}$ i $\beta = \angle{CBA}$. Neka pravac $CS$ siječe pravac $AB$ u točki $D$ koja se nalazi između točaka $A$ i $B$. Dokaži da vrijedi $$ \frac{\left\vert SD \right\vert}{\left\vert SC \right\vert} = \left\vert \frac{\cos\left(\alpha + \beta\right)}{\cos\left(\alpha-\beta\right)}\right\vert \text{.} $$