IMO Shortlist 2006 problem G6
Dodao/la:
arhiva2. travnja 2012. Circles
![w_{1}](/media/m/3/4/e/34e5d85837247f4553674a9da9c4c426.png)
and
![w_{2}](/media/m/6/a/5/6a5bee98427f6ba6dc5934c061057d06.png)
with centres
![O_{1}](/media/m/a/4/0/a4032d61b59d798b9dac58cad84150cd.png)
and
![O_{2}](/media/m/c/7/0/c7092dc3959c183ed68e2e991c40970e.png)
are externally tangent at point
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
and internally tangent to a circle
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
at points
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
respectively. Line
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
is the common tangent of
![w_{1}](/media/m/3/4/e/34e5d85837247f4553674a9da9c4c426.png)
and
![w_{2}](/media/m/6/a/5/6a5bee98427f6ba6dc5934c061057d06.png)
at
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
. Let
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
be the diameter of
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
perpendicular to
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
, so that
![A, E, O_{1}](/media/m/0/8/6/0866ffca0664abb98c19c8cf9aa3dfeb.png)
are on the same side of
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
. Prove that lines
![AO_{1}](/media/m/3/6/1/3617502a3e02b17a2b3e80802b72c36a.png)
,
![BO_{2}](/media/m/5/9/e/59e9a8185f63de6d05bf646f560bf617.png)
,
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
and
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
are concurrent.
%V0
Circles $w_{1}$ and $w_{2}$ with centres $O_{1}$ and $O_{2}$ are externally tangent at point $D$ and internally tangent to a circle $w$ at points $E$ and $F$ respectively. Line $t$ is the common tangent of $w_{1}$ and $w_{2}$ at $D$. Let $AB$ be the diameter of $w$ perpendicular to $t$, so that $A, E, O_{1}$ are on the same side of $t$. Prove that lines $AO_{1}$, $BO_{2}$, $EF$ and $t$ are concurrent.
Izvor: Međunarodna matematička olimpijada, shortlist 2006