2004 Indonesia MO Problems/Problem 4
Problem
There exists 4 circles, , such that
is tangent to both
and
,
is tangent to both
and
,
is both tangent to
and
, and
is both tangent to
and
. Show that all these tangent points are located on a circle.
Solution
Let be the centers of circle
and
respectively. Also, let the tangent point of circles
and
be
, let the tangent point of circles
and
be
, let the tangent point of circles
and
be
, and let the tangent point of circles
and
be
. Finally, let
be
and
respectively.
Note that are isosceles triangles, so
degrees,
degrees,
degrees, and
degrees. The sum of the angles in a line is
so
degrees and
degrees.
That means degrees. Since
is a quadrilateral,
so
Thus, all the tangent points of the four circles can be located in another circle.
See Also
2004 Indonesia MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 5 |
All Indonesia MO Problems and Solutions |