IMO Shortlist 1998 problem G8
Kvaliteta:
Avg: 0,0Težina:
Avg: 9,0 Let
be a triangle such that
and
. The tangent at
to the circumcircle
of triangle
meets the line
at
. Let
be the reflection of
in the line
, let
be the foot of the perpendicular from
to
, and let
be the midpoint of the segment
. Let the line
intersect the circle
again at
.
Prove that the line
is tangent to the circumcircle of triangle
.
commentEdited by Orl.
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![\angle A=90^{\circ }](/media/m/f/c/3/fc34d22f02703a6423061de6e40539c1.png)
![\angle B<\angle C](/media/m/3/4/e/34e1c87e8d93768803cb82e4dbbb7ebf.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![BE](/media/m/e/e/2/ee25cd134664bc0c8d7fdbba81e54f90.png)
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
![AX](/media/m/3/a/8/3a8b3cfe621304b5621fb712075419c2.png)
![BY](/media/m/c/e/b/ceb91029af823458377562b596edba2f.png)
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
Prove that the line
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
![ADZ](/media/m/c/4/3/c430563874ba91e08a5969e6cc32e7cf.png)
commentEdited by Orl.
Izvor: Međunarodna matematička olimpijada, shortlist 1998