Državno natjecanje 2003 SŠ3 1
Dodao/la:
arhiva1. travnja 2012. U trokutu
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
je
![a = |BC|](/media/m/2/e/2/2e297ad4b5dcf6b96eaf6dfa5c0d065b.png)
,
![b = |CA|](/media/m/e/9/0/e900c7bd1191ec9811e9c11fab3b546c.png)
,
![c = |AB|](/media/m/d/9/f/d9f121433eb5aad11fa2e12d162c1090.png)
,
![\alpha = \angle CAB](/media/m/e/4/f/e4fa28be1d3e2d4c9421da1ff78814ec.png)
,
![\beta = \angle ABC](/media/m/f/3/1/f317ffeca7b400883e507d522e3f6334.png)
,
![\gamma = \angle BAC](/media/m/a/2/5/a259b052c82b09c58471e48a3ebbe6ae.png)
.
![a)](/media/m/f/0/8/f0844437a160b45486aedcc02b92949d.png)
Ako je
![\alpha = 3\beta](/media/m/b/d/e/bde876e5cdd2cf6bb1b09622d817b0ee.png)
, dokažite da je
![(a^2 - b^2)(a - b) = bc^2](/media/m/7/3/5/7353db4bac186a421166ed5381753c8f.png)
.
![b)](/media/m/d/2/f/d2f292cd6a69e9158afe71ba9d830da4.png)
Vrijedi li obrat? Obrazložite.
%V0
U trokutu $ABC$ je $a = |BC|$, $b = |CA|$, $c = |AB|$, $\alpha = \angle CAB$, $\beta = \angle ABC$, $\gamma = \angle BAC$.
$a)$ Ako je $\alpha = 3\beta$, dokažite da je $(a^2 - b^2)(a - b) = bc^2$.
$b)$ Vrijedi li obrat? Obrazložite.
Izvor: Državno natjecanje iz matematike 2003