Problem 2

a

b

, and

c

are the sides and

\alpha

\beta

, and

\gamma

are the respective angles of a triangle for which

a+b=\tan\frac{\gamma}{2}\left(a\tan\alpha+b\tan\beta\right)

, prove that the triangle is isosceles.

Step 6 of 6: Conclude the triangle is isosceles

In plain words

Once the angle equality is forced in every case, converting it back to a side equality via the law of sines finishes the proof.

\alpha=\beta \ \Longrightarrow\ a=b

Detailed analysis

Since $\alpha=\beta$ and the sides opposite equal angles are equal ( $a=2R\sin\alpha=2R\sin\beta=b$ ), triangle $ABC$ is isosceles with $a=b$ .