MEMO 2013 ekipno problem 6
Kvaliteta:
Avg: 0,0Težina:
Avg: 6,0 Neka je točka
unutar šiljastokutnog trokuta
takva da je pravac
zajednička tangenta kružnica opisanih trokutima
i
. Neka je točka
sjecište pravaca
i
te neka je točka
sjecište pravaca
i
. Neka je točka
sjecište pravca
i simetrale dužine
. Opisana kružnica trokuta
i kružnica
polumjera
sa središtem u točki
sijeku se u točkama
i
.
Dokažite da je dužina
promjer kružnice
.
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![AKB](/media/m/f/d/d/fdd6903797073c8a2ca2559858202f09.png)
![AKC](/media/m/6/b/5/6b56cfcf11045a55cdf3a579fa563451.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![CK](/media/m/e/b/0/eb0d71f0932c3a96b200928cd07df5f0.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![BK](/media/m/b/9/c/b9cf4d2548fb2edc8dc197004ed51cf4.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![\overline{DE}](/media/m/f/f/1/ff1151be8c83f905b39371ec47cb7144.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![\overline{FD}](/media/m/d/5/8/d581e3fd3815057766db5a52e8b3c88e.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
Dokažite da je dužina
![\overline{PQ}](/media/m/f/5/d/f5d8cee1cf25e485433f6a207a20ad45.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
Izvor: Srednjoeuropska matematička olimpijada 2013, ekipno natjecanje, problem 6