Vrhovi kocke u prostornom koordinatnom sustavu s ishodištem
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
su u točkama
![A(1,1,1)](/media/m/7/6/4/7645f54697ba909e3b1bd4dfa36b1d7e.png)
,
![A^\prime(-1,-1,-1)](/media/m/f/4/9/f49af25d504e4f73a2de26e42493e1ff.png)
,
![B(-1,1,1)](/media/m/6/7/b/67bbc0ea7687261a57fbc032675afcde.png)
,
![B^\prime(1,-1,-1)](/media/m/c/8/c/c8cee6a4e4b6ccf44dc6a7787c641f4b.png)
,
![C(-1,-1,1)](/media/m/4/3/2/43234ec24ca9a676a673bffaa520711f.png)
,
![C^\prime(1,1,-1)](/media/m/9/1/5/9155955a862e47c7acf908345dd9fb8a.png)
,
![D(1,-1,1)](/media/m/c/c/4/cc4d8d418c012d3c09f575f7957734a5.png)
,
![D^\prime(-1,1,-1)](/media/m/2/e/4/2e4cb335f8f198a346ad699350c815e4.png)
. Točka
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
je središte kocki opisane sfere. Neka točka
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
nije na toj sferi i
![d=|OT|](/media/m/6/f/6/6f6898897a47602876f932d073e85944.png)
. Označimo s
![\alpha = \angle ATA^\prime](/media/m/8/f/0/8f01beadc6b795ee473314712aae3bdb.png)
,
![\beta = \angle BTB^\prime](/media/m/4/5/3/4531f1f413560f98abbaa39f0fd3356d.png)
,
![\gamma = \angle CTC^\prime](/media/m/1/e/a/1ea1dba03efc9ad36a161d9f137060e7.png)
,
![\delta = \angle DTD^\prime](/media/m/a/6/c/a6c9a9001ba0921c81578f058f6fd71c.png)
. Dokažite da je
%V0
Vrhovi kocke u prostornom koordinatnom sustavu s ishodištem $O$ su u točkama $A(1,1,1)$, $A^\prime(-1,-1,-1)$, $B(-1,1,1)$, $B^\prime(1,-1,-1)$, $C(-1,-1,1)$, $C^\prime(1,1,-1)$, $D(1,-1,1)$, $D^\prime(-1,1,-1)$. Točka $O$ je središte kocki opisane sfere. Neka točka $T$ nije na toj sferi i $d=|OT|$. Označimo s $\alpha = \angle ATA^\prime$, $\beta = \angle BTB^\prime$, $\gamma = \angle CTC^\prime$, $\delta = \angle DTD^\prime$. Dokažite da je
$$tg^2 \alpha + tg^2 \beta + tg^2 \gamma + tg^2 \delta = \frac{32d^2}{(d^2-3)^2}.$$