2010 AIME II Problems/Problem 15
Problem 15
In triangle ,
,
, and
. Points
and
lie on
with
and
. Points
and
lie on
B with
and
. Let
be the point, other than
, of intersection of the circumcircles of
and
. Ray
meets
at
. The ratio
can be written in the form
, where
and
are relatively prime positive integers. Find
.
Solution
Let .
since
. Since quadrilateral
is cyclic,
and
, yielding
and
. Multiplying these together yields
.
. Also,
is the center of spiral similarity of segments
and
, so
. Therefore,
, which can easily be computed by the angle bisector theorem to be
. It follows that
, giving us an answer of
.
Note: Spiral similarities may sound complex, but they're really not. The fact that is really just a result of simple angle chasing.
Source: [1] by Zhero
See Also
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