Consider a sequence of circles
![K_1,K_2,K_3,K_4, \ldots](/media/m/c/4/a/c4ad7646ec9d3750e66e67891e0bc43c.png)
of radii
![r_1, r_2, r_3, r_4, \ldots](/media/m/3/0/3/303fe66765ed2aa5dd7d5a2e867ffc85.png)
, respectively, situated inside a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. The circle
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
is tangent to
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
;
![K_2](/media/m/3/1/d/31d4e457d9fd74913d14bd19a565ce00.png)
is tangent to
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
,
![BA](/media/m/f/3/e/f3ee5efe9b25bd27cd3ada1235d36017.png)
, and
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
;
![K_3](/media/m/d/7/c/d7c055636fbf1f3b807bc72796996893.png)
is tangent to
![K_2](/media/m/3/1/d/31d4e457d9fd74913d14bd19a565ce00.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
, and
![CB](/media/m/d/d/7/dd7ebd02df3c940a7e47b8a09480e1b1.png)
;
![K_4](/media/m/0/9/2/092fbf53f6b5c75e84a2296a13963fa2.png)
is tangent to
![K_3](/media/m/d/7/c/d7c055636fbf1f3b807bc72796996893.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
; etc.
(a) Prove the relation
![r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right)](/media/m/0/4/0/04060c38cdd9045e940f5468fb441cb7.png)
where
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
is the radius of the incircle of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Deduce the existence of a
![t_1](/media/m/d/3/3/d332512f256ff456647ffb3819f44831.png)
such that
![r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1](/media/m/2/3/1/231119a45e1d2384e3e9a9d1b72b635a.png)
(b) Prove that the sequence of circles
![K_1,K_2, \ldots](/media/m/c/6/d/c6d12024b0076a265334bddbb3e28f00.png)
is periodic.
%V0
Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc.
(a) Prove the relation
$$r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right)$$
where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that
$$r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1$$
(b) Prove that the sequence of circles $K_1,K_2, \ldots$ is periodic.