IMO Shortlist 1966 problem 59
Dodao/la:
arhiva2. travnja 2012. Let
![a,b,c](/media/m/3/6/4/36454fdb50fc50f021324b33a6b513e3.png)
be the lengths of the sides of a triangle, and
![\alpha, \beta, \gamma](/media/m/2/8/6/286052c3ef6a627f9d4f5349ddaf2ba7.png)
respectively, the angles opposite these sides. Prove that if
![a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta})](/media/m/0/2/6/026e1f442cd6b79ea4ea373ad7ffc764.png)
the triangle is isosceles.
%V0
Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if $$a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta})$$ the triangle is isosceles.
Izvor: Međunarodna matematička olimpijada, shortlist 1966