2023 AIME II Problems/Problem 12
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
.
Solution 2 (
Define to be the foot of the altitude from
to
. Furthermore, define
to be the foot of the altitude from
to
. From here, one can find
, either using the 13-14-15 triangle or by calculating the area of
two ways. Then, we find
and
using Pythagorean theorem. Let
. By AA similarity,
and
are similar. By similarity ratios,
Thus,
. Similarly,
. Now, we angle chase from our requirement to obtain new information.
Take the tangent of both sides to obtain
By the definition of the tangent function on right triangles, we have
,
, and
. By abusing the tangent angle addition formula, we can find that
By substituting
,
and using tangent angle subtraction formula we find that
Finally, using similarity formulas, we can find
. Plugging in
and
, we find that
Thus, our final answer is
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)