2023 AIME II Problems/Problem 12
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Solution
Because is the midpoint of
, following from the Steward's theorem,
.
Because ,
,
,
are concyclic,
,
.
Denote .
In , following from the law of sines,
Thus,
In , following from the law of sines,
Thus,
Taking , we get
In , following from the law of sines,
Thus, Equations (2) and (3) imply
Next, we compute and
.
We have
We have
Taking (5) and (6) into (4), we get .
Therefore, the answer is
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)