IMO Shortlist 1966 problem 32
Dodao/la:
arhiva2. travnja 2012. The side lengths
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
of a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
form an arithmetical progression (such that
![b-a=c-b](/media/m/e/8/d/e8d14530ef7d5299531992f80958ba86.png)
). The side lengths
![c_{1}](/media/m/c/3/b/c3b499e0dedfead7a0ba6b740fdb803e.png)
of a triangle
![A_{1}B_{1}C_{1}](/media/m/7/1/2/712cbb7e7d655a0abf51f5e564137450.png)
also form an arithmetical progression (with
![b_{1}-a_{1}=c_{1}-b_{1}](/media/m/4/a/0/4a0d678dd542505a6eba09f08bdac6d7.png)
). (Hereby,
![a=BC](/media/m/3/4/7/3474740afea35293a1c3b7487f6bbb52.png)
,
![b=CA](/media/m/7/0/5/705fdf372ddf86a2bdb21eee6c2b98e7.png)
,
![c=AB](/media/m/e/3/0/e30285924bc1a71265c9bd7beb5f3ea5.png)
,
![a_{1}=B_{1}C_{1}](/media/m/a/f/0/af0bb9b07b59ab7862d3dbe9e346b739.png)
,
![b_{1}=C_{1}A_{1}](/media/m/7/a/0/7a044df5e9a2ca230fca201ed282f002.png)
,
![c_{1}=A_{1}B_{1}](/media/m/1/8/f/18f7f8816d5c39931275b510039e3a99.png)
.) Moreover, we know that
Show that triangles
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
and
![A_{1}B_{1}C_{1}](/media/m/7/1/2/712cbb7e7d655a0abf51f5e564137450.png)
are similar.
%V0
The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). (Hereby, $a=BC$, $b=CA$, $c=AB$, $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$.) Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$
Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.
Izvor: Međunarodna matematička olimpijada, shortlist 1966