Difference between revisions of "1970 IMO Problems/Problem 1"
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* [[1970 IMO Problems]] | * [[1970 IMO Problems]] | ||
− | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=366686#p366686 | + | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=366686#p366686 Discussion on AoPS/Mathlinks] |
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] |
Revision as of 18:21, 18 September 2006
Problem
( Proposed by Poland ) Let be a point on the side
of
. Let
, and
be the inscribed circles of triangles
, and
. Let
, and
be the radii of the exscribed circles of the same triangles that lie in the angle
. Prove that
.
Solution
We use the conventional triangle notations.
Let be the incenter of
, and let
be its excenter to side
. We observe that
,
and likewise,
Simplifying the quotient of these expressions, we obtain the result
.
Thus we wish to prove that
.
But this follows from the fact that the angles and
are supplementary.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.