Neka je
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
pravokutni trokut s katetama duljina
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
i
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
i hipotenuzom duljine
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
, a
![\angle BCA=90^\circ](/media/m/1/e/5/1e5ca90ca75e85f3bdfc27a0695c2f24.png)
. Neka je
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
tom trokutu opisana kružnica,
![k_1](/media/m/3/5/6/35656cbf3adb55dddd30996fc068363b.png)
kružnica koja dodiruje hipotenuzu, visinu
![\overline{CD}](/media/m/3/3/8/338870e40f3ea7992d83158230115a5f.png)
i luk
![\widehat{BC}](/media/m/8/a/9/8a93a5cf4249316301d29f3ec352e014.png)
kružnice
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
; te
![k_2](/media/m/6/a/b/6abbe24dbf6713b55498fe55ab050d06.png)
kružnica koja dodiruje hipotenuzu, visinu
![\overline{CD}](/media/m/3/3/8/338870e40f3ea7992d83158230115a5f.png)
i luk
![\widehat{AC}](/media/m/c/4/7/c4757f6b9ac6c0dc3a423db640e52815.png)
kružnice
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
. Ako su
![r_1](/media/m/9/0/1/901ecb943995b3585cd44466e1b750cb.png)
i
![r_2](/media/m/9/0/6/90608ee2be6d3b5c7f96a6ca45780ec4.png)
polumjeri kružnica
![k_1](/media/m/3/5/6/35656cbf3adb55dddd30996fc068363b.png)
i
![k_2](/media/m/6/a/b/6abbe24dbf6713b55498fe55ab050d06.png)
dokažite da je
%V0
Neka je $ABC$ pravokutni trokut s katetama duljina $a$ i $b$ i hipotenuzom duljine $c$, a $\angle BCA=90^\circ$. Neka je $k$ tom trokutu opisana kružnica, $k_1$ kružnica koja dodiruje hipotenuzu, visinu $\overline{CD}$ i luk $\widehat{BC}$ kružnice $k$; te $k_2$ kružnica koja dodiruje hipotenuzu, visinu $\overline{CD}$ i luk $\widehat{AC}$ kružnice $k$. Ako su $r_1$ i $r_2$ polumjeri kružnica $k_1$ i $k_2$ dokažite da je $$
r_1+r_2=a+b-c.
$$