Upisana kružnica šiljastokutnog trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
dodiruje stranice
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
,
![\overline{CA}](/media/m/c/e/9/ce9fb8497710464615e1d00d148c5663.png)
i
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
redom u točkama
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
i
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
. Središte te kružnice je točka
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
, a pravac
![DS](/media/m/0/0/a/00ac1134d3e27b445b8edd53a125e06e.png)
siječe dužinu
![\overline{EF}](/media/m/7/3/6/736526ec2c1c20572842175dc3523f2c.png)
u točki
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
. Ako je
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
polovište stranice
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
, dokaži da su točke
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
i
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
kolinearne.
%V0
Upisana kružnica šiljastokutnog trokuta $ABC$ dodiruje stranice $\overline{BC}$, $\overline{CA}$ i $\overline{AB}$ redom u točkama $D$, $E$ i $F$. Središte te kružnice je točka $S$, a pravac $DS$ siječe dužinu $\overline{EF}$ u točki $P$. Ako je $M$ polovište stranice $\overline{BC}$, dokaži da su točke $A$, $P$ i $M$ kolinearne.