2021 AIME I Problems/Problem 13
Contents
Problem
Circles and
with radii
and
, respectively, intersect at distinct points
and
. A third circle
is externally tangent to both
and
. Suppose line
intersects
at two points
and
such that the measure of minor arc
is
. Find the distance between the centers of
and
.
Solution 1 (Properties of Radical Axis)
Let and
be the center and radius of
, and let
and
be the center and radius of
.
Since extends to an arc with arc
, the distance from
to
is
. Let
. Consider
. The line
is perpendicular to
and passes through
. Let
be the foot from
to
; so
. We have by tangency
and
. Let
.
Since
is on the radical axis of
and
, it has equal power with respect to both circles, so
since
. Now we can solve for
and
, and in particular,
We want to solve for
. By the Pythagorean Theorem (twice):
Therefore,
.
Solution 2 (Linearity)
Let and
be the centers of
and
respectively, and let
be the center of
. Then, the distance from
to the radical axis
of
is equal to
. Let
and
the orthogonal projection of
onto line
. Define the function
by
Then
By Linearity of Power of a Point,
Notice that
and
, thus
since
is nonzero.
Solution 3
Denote by ,
, and
the centers of
,
, and
, respectively. Let
and
denote the radii of
and
respectively,
be the radius of
, and
the distance from
to the line
. We claim that
where
. This solves the problem, for then the
condition implies
, and then we can solve to get
.
Denote by and
the centers of
and
respectively. Set
as the projection of
onto
, and denote by
the intersection of
with
. Note that
. Now recall that
Furthermore, note that
Substituting the first equality into the second one and subtracting yields
which rearranges to the desired.
Solution 4 (Quick)
Suppose we label the points as shown here. By radical axis, the tangents to at
and
intersect on
. Thus
is harmonic, so the tangents to
at
and
intersect at
. Moreover,
because both
and
are perpendicular to
, and
because
. Thus
by similar triangles.
~mathman3880
Video Solution
Who wanted to see animated video solutions can see this. I found this really helpful.
P.S: This video is not made by me. And solution is same like below solutions.
≈@rounak138
See Also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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