Difference between revisions of "2009 USAMO Problems/Problem 1"
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Revision as of 12:44, 4 July 2013
Problem
Given circles and
intersecting at points
and
, let
be a line through the center of
intersecting
at points
and
and let
be a line through the center of
intersecting
at points
and
. Prove that if
and
lie on a circle then the center of this circle lies on line
.
Solution
Let be the circumcircle of
. Define
to be the radius and
to be the center of the circle
. Then
lies on the line passing through the intersections of
, or their radical axis, and similarly
lies on the radical axis of
. Then, the power of
with respect to
are the same, and similarly for
:
Subtracting gives
, so
lies on the radical axis of
. Thus
are collinear.
See also
2009 USAMO (Problems • Resources) | ||
Preceded by First question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.