1970 IMO Problems/Problem 1
Revision as of 18:21, 18 September 2006 by Boy Soprano II (talk | contribs) (→Resources: corrected typo)
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.