Točno
23. listopada 2013. 18:08 (10 godine, 8 mjeseci)
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Baza
![n=3](/media/m/6/e/8/6e8cc663572ec564892ed13a28debcb1.png)
Trokut ima
dijagonala, a ![\dfrac{3(3-3)}{2}=\dfrac{3\cdot 0}{2} = 0](/media/m/c/c/b/ccb51b589a7c4dfc7ee0755cb9c1d088.png)
Dakle, baza je dokazana.
Pretpostavka
-terokut ima tocno
dijagonala.
Korak
Zelimo dokazati da
-terkokut ima
dijagonala.
Oznacimo neka tri uzastopna vrha tog
-terokuta sa
,
, i
, tako da je
izmedu
i
. Sada povucimo duzinu
.
Dobili smo
-terokut (iz
-terokuta smo izbacili jedan vrh, konkretno
) i trokut
. U
-teroktu, po pretpostavci, ima
dijagonala. U trokutu ih ima
.
Sve dijagonale
-terokuta koje jos nismo brojali zavrsavaju u tocki
. Takvih dijagonala ima
(
mozemo povezati sa svim vrhovima osim sa njime samim, a sa svoja dva susjedna vrha daje stranice, a ne dijagonale). I jos nismo brojali duzinu
, koja je u
-terokutu stranica, ali u
-terokutu je dijagonala.
Dakle, ukupno imamo
dijagonala iz
-terokuta,
dijagonale iz tocke
i dijagonalu
, sto je u sumi
![\dfrac{n(n-3)}{2}+(n-2)+1](/media/m/3/0/4/304ae7df887081cd056fe7940dd1cf72.png)
![= \dfrac{n(n-3)+2(n-1)}{2}](/media/m/a/e/c/aec3ae0dd25c5f5794d196e68abcfac2.png)
![n=3](/media/m/6/e/8/6e8cc663572ec564892ed13a28debcb1.png)
Trokut ima
![0](/media/m/7/b/8/7b8b0b058cf5852d38ded7a42d6292f5.png)
![\dfrac{3(3-3)}{2}=\dfrac{3\cdot 0}{2} = 0](/media/m/c/c/b/ccb51b589a7c4dfc7ee0755cb9c1d088.png)
Dakle, baza je dokazana.
Pretpostavka
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![\dfrac{n(n-3)}{2}](/media/m/6/1/6/616d6a064e37650a6ff61b2dc690995d.png)
Korak
Zelimo dokazati da
![(n+1)](/media/m/3/a/9/3a9b2b28796496249e968c81a438d4da.png)
![\dfrac{(n+1)(n-2)}{2}](/media/m/8/a/c/8ac21791571e79508a7ad461ca9b1ec3.png)
Oznacimo neka tri uzastopna vrha tog
![(n+1)](/media/m/3/a/9/3a9b2b28796496249e968c81a438d4da.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
Dobili smo
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![(n+1)](/media/m/3/a/9/3a9b2b28796496249e968c81a438d4da.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![\dfrac{n(n-3)}{2}](/media/m/6/1/6/616d6a064e37650a6ff61b2dc690995d.png)
![0](/media/m/7/b/8/7b8b0b058cf5852d38ded7a42d6292f5.png)
Sve dijagonale
![(n+1)](/media/m/3/a/9/3a9b2b28796496249e968c81a438d4da.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![((n+1)-3)](/media/m/5/1/c/51c0d1826246f1aa3b59a42c8129233f.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![(n+1)](/media/m/3/a/9/3a9b2b28796496249e968c81a438d4da.png)
Dakle, ukupno imamo
![\dfrac{n(n-3)}{2}](/media/m/6/1/6/616d6a064e37650a6ff61b2dc690995d.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![(n-2)](/media/m/7/c/0/7c04b504e3e3cf329fe94cb0034e0c5d.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![\dfrac{n(n-3)}{2}+(n-2)+1](/media/m/3/0/4/304ae7df887081cd056fe7940dd1cf72.png)
![= \dfrac{n(n-3)+2(n-1)}{2}](/media/m/a/e/c/aec3ae0dd25c5f5794d196e68abcfac2.png)
![=\dfrac{(n+1)(n-2)}{2}](/media/m/d/b/2/db2e902408cfcc905771b024d23aacb8.png)