Difference between revisions of "1970 IMO Problems/Problem 1"
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Revision as of 00:14, 14 November 2024
Contents
[hide]Problem
Let be a point on the side
of
. Let
, and
be the inscribed circles of triangles
, and
. Let
, and
be the radii of the escribed 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.
Solution 2
By similar triangles and the fact that both centers lie on the angle bisector of , we have
, where
is the semi-perimeter of
. Let
have sides
, and let
. After simple computations, we see that the condition, whose equivalent form is
is also equivalent to Stewart's Theorem (see Stewart's_theorem or https://en.wikipedia.org/wiki/Stewart's_theorem)
Solution 3
[COMING SOON.]
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
1970 IMO (Problems) • Resources | ||
Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |