neka je
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
tocka na simetrali kuta
![\angle BAC](/media/m/b/2/1/b21a9e466104c5d33646432221e142be.png)
trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, a
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
i
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
redom tocke na stranicama
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
i
![\overline{AC}](/media/m/d/9/5/d95354f0f833a5fda9c16a01a878c14f.png)
, takve da je
![\angle ABI = \angle NIC](/media/m/4/5/7/4579f96ffcec4c6328adaff001167c2c.png)
i
![\angle ACI = \angle MIB](/media/m/f/f/6/ff6bd14903c349ce83b7b6d6315e1f03.png)
. dokazite da je
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
srediste upisane kruznice trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
ako i samo ako su tocke
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
,
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
i
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
kolinearne.
%V0
neka je $I$ tocka na simetrali kuta $\angle BAC$ trokuta $ABC$, a $M$ i $N$ redom tocke na stranicama $\overline{AB}$ i $\overline{AC}$, takve da je $\angle ABI = \angle NIC$ i $\angle ACI = \angle MIB$. dokazite da je $I$ srediste upisane kruznice trokuta $ABC$ ako i samo ako su tocke $M$, $N$ i $I$ kolinearne.