IMO Shortlist 1994 problem G1
Dodao/la:
arhiva2. travnja 2012. ![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
are points on a semicircle. The tangent at
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
meets the extended diameter of the semicircle at
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
, and the tangent at
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
meets it at
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
, so that
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
are on opposite sides of the center. The lines
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
meet at
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
.
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
is the foot of the perpendicular from
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
to
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
. Show that
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
bisects angle
%V0
$C$ and $D$ are points on a semicircle. The tangent at $C$ meets the extended diameter of the semicircle at $B$, and the tangent at $D$ meets it at $A$, so that $A$ and $B$ are on opposite sides of the center. The lines $AC$ and $BD$ meet at $E$. $F$ is the foot of the perpendicular from $E$ to $AB$. Show that $EF$ bisects angle $CFD$
Izvor: Međunarodna matematička olimpijada, shortlist 1994