Državno natjecanje 2011 SŠ3 3
Dodao/la:
arhiva1. travnja 2012. U trokutu
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
vrijedi
![\left\vert AB \right\vert = \left\vert AC \right\vert](/media/m/6/7/4/674f17e6e38bc7de472c2536bef08494.png)
. Na stranici
![\overline{AC}](/media/m/d/9/5/d95354f0f833a5fda9c16a01a878c14f.png)
nalazi se točka
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
takva da je
![\left\vert AD \right\vert < \left\vert CD \right\vert](/media/m/8/2/9/829d505383f117d64496975bc028e857.png)
, a na dužini
![\overline{BD}](/media/m/7/3/2/732e8894e57eb20026de06c47885ae55.png)
točka
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
takva da je
![\angle{APC}](/media/m/1/b/3/1b33b3d244a74f6c53d5af70bb66f3d7.png)
pravi kut. Ako je
![\angle{ABP} = \angle{BCP}](/media/m/9/3/4/9348d7f21f9ba1ddd2b7ce237eecedfe.png)
, odredi
![\left\vert AD \right\vert : \left\vert CD \right\vert](/media/m/c/9/7/c975f62c28515b4be6842c9fa03dee1f.png)
.
%V0
U trokutu $ABC$ vrijedi $\left\vert AB \right\vert = \left\vert AC \right\vert$. Na stranici $\overline{AC}$ nalazi se točka $D$ takva da je $\left\vert AD \right\vert < \left\vert CD \right\vert$, a na dužini $\overline{BD}$ točka $P$ takva da je $\angle{APC}$ pravi kut. Ako je $\angle{ABP} = \angle{BCP}$, odredi $\left\vert AD \right\vert : \left\vert CD \right\vert$.
Izvor: Državno natjecanje iz matematike 2011