Difference between revisions of "2012 USAJMO Problems/Problem 1"
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Revision as of 18:03, 3 July 2013
Problem
Given a triangle , let
and
be points on segments
and
, respectively, such that
. Let
and
be distinct points on segment
such that
lies between
and
,
, and
. Prove that
,
,
,
are concyclic (in other words, these four points lie on a circle).
Solution
Since , the circumcircle of triangle
is tangent to
at
. Similarly, since
, the circumcircle of triangle
is tangent to
at
.
For the sake of contradiction, suppose that the circumcircles of triangles and
are not the same circle. Since
,
lies on the radical axis of both circles. However, both circles pass through
and
, so the radical axis of both circles is
. Hence,
lies on
, which is a contradiction.
Therefore, the two circumcircles are the same circle. In other words, ,
,
, and
all lie on the same circle.
See also
2012 USAJMO (Problems • Resources) | ||
First Problem | Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.