Dan je trokut
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
i točka
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
unutar njega. Pravac paralelan s
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
koji prolazi kroz
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
siječe stranice
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
i
![\overline{CA}](/media/m/c/e/9/ce9fb8497710464615e1d00d148c5663.png)
u točkama
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
i
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
redom. Pravac paralelan s
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
koji prolazi kroz
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
siječe stranice
![\overline{CA}](/media/m/c/e/9/ce9fb8497710464615e1d00d148c5663.png)
i
![\overline{BA}](/media/m/8/e/4/8e4ee9123d5b4f2856ea0a300dbd1186.png)
u točkama
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
i
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
redom, dok pravac paralelan s
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
koji prolazi kroz
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
siječe stranice
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
i
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
u točkama
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
i
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
redom. Dokaži da vrijedi
![(PDBL) \cdot (PECM) \cdot (PFAN) = 8 \cdot (PFM) \cdot (PEL) \cdot (PDN) \text{,}](/media/m/c/d/2/cd2d733ccde02b217532ad3010f9a335.png)
gdje
![(XYZ)](/media/m/3/d/7/3d7184affbe4169bb64970478233616d.png)
i
![(XYZT)](/media/m/d/0/2/d025e719596c58a582e6b9ee6333b4ab.png)
označavaju površinu trokuta
![XYZ](/media/m/1/3/d/13dab5022dd1d33f3d299852f2f54cfb.png)
, odnosno površinu četverokuta
![XYZT](/media/m/2/b/1/2b18e476277add9bfcf49231d4adf921.png)
.
%V0
Dan je trokut $ABC$ i točka $P$ unutar njega. Pravac paralelan s $AB$ koji prolazi kroz $P$ siječe stranice $\overline{BC}$ i $\overline{CA}$ u točkama $L$ i $F$ redom. Pravac paralelan s $BC$ koji prolazi kroz $P$ siječe stranice $\overline{CA}$ i $\overline{BA}$ u točkama $M$ i $D$ redom, dok pravac paralelan s $CA$ koji prolazi kroz $P$ siječe stranice $\overline{AB}$ i $\overline{BC}$ u točkama $N$ i $E$ redom. Dokaži da vrijedi $$
(PDBL) \cdot (PECM) \cdot (PFAN) = 8 \cdot (PFM) \cdot (PEL) \cdot (PDN) \text{,}
$$ gdje $(XYZ)$ i $(XYZT)$ označavaju površinu trokuta $XYZ$, odnosno površinu četverokuta $XYZT$.