IMO Shortlist 1990 problem 11
Dodao/la:
arhiva2. travnja 2012. Chords
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
of a circle intersect at a point
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
inside the circle. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be an interior point of the segment
![EB](/media/m/2/7/f/27fbd615d52646083fc755b020aecb89.png)
. The tangent line at
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
to the circle through
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
, and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
intersects the lines
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
at
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
and
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
, respectively. If
![\frac {AM}{AB} = t,](/media/m/6/1/c/61c24221528d5379575b142038ed858f.png)
find
![\frac {EG}{EF}](/media/m/7/d/e/7deb064024e6a6de0d3771ef122cb328.png)
in terms of
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
.
%V0
Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$. The tangent line at $E$ to the circle through $D$, $E$, and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$, respectively. If
$$\frac {AM}{AB} = t,$$
find $\frac {EG}{EF}$ in terms of $t$.
Izvor: Međunarodna matematička olimpijada, shortlist 1990