1996 USAMO Problems/Problem 5
Problem
Let be a triangle, and
an interior point such that
,
,
and
. Prove that the triangle is isosceles.
Solution 1
Clearly, and
. Now by the Law of Sines on triangles
and
, we have
and
Combining these equations gives us
Without loss of generality, let
and
. Then by the Law of Cosines, we have
Thus, , our desired conclusion.
Solution 2
By the law of sines, and
, so
.
Let . Then,
. By the law of sines,
.
So, we have .
First, let's focus on . By the identities
and
, we have
Substituting back in to the original equality, and using the identity and the facts that
and
, we have
Therefore, . Then, using the identity
,
The only acute angle satisfying this equality is . Therefore,
and
. Thus,
is isosceles.
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