2009 USAMO Problems/Problem 1
Contents
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 1
Let be the circumcircle of
,
to be the radius of
, and
to be the center of the circle
, where
. Note that
and
are the radical axises of
,
and
,
respectively. Hence, by power of a point(the power of
can be expressed using circle
and
and the power of
can be expressed using circle
and
),
Subtracting these two equations yields that
, so
must lie on the radical axis of
,
.
~AopsUser101
Solution 2
Define and
similarly to above. Note that
is perpendicular to
and
is perpendicular to
. Thus, the intersection of
and
must be the orthocenter of triangle
. Define this as point
. Extending line
to meet
, we note that
is perpendicular to
.
In addition, note that by the radical axis theorem, the intersection of and
must also lie on the radical axis of
and
. Because the radical axis of
and
is perpendicular to
and contains
, it must also contain
, and we are done.
It seems like. But the radical axis theorem has its own conditions. Consider a very special when all centers 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 |
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