2016 AIME I Problems/Problem 15
Problem
Circles and
intersect at points
and
. Line
is tangent to
and
at
and
, respectively, with line
closer to point
than to
. Circle
passes through
and
intersecting
again at
and intersecting
again at
. The three points
,
,
are collinear,
,
, and
. Find
.
Solution
By radical axis theorem concur at point
.
Let and
intersect at
. Note that because
and
are cyclic, by Miquel theorem
are cyclic as well. Thus
and
Thus
and
so
is a parallelogram. Hence
and
. But notice that
and
are similar by
Similarity, so
. But
Hence
See Also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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