Determine the elements of a triangle knowing relationships between lengths and angles A triangle's sides' lengths are three successive numbers (meaning b=a+1, c=a+2). The smallest angle is a half of the triangle's biggest angle.
Find the area of this triangle and its angles. 
Solution: 
Sides are 4, 5 and 6.
Area: $15\sqrt{7}/4$
Angles : $41° 24' 34"$ and $82°49'8"$

Hint:
Solve
$$\frac{\sin\theta}{a}=\frac{\sin3\theta}{a+1}=\frac{\sin2\theta}{a+2}.$$


$$\frac{\sin\theta}a=\frac{4\sin\theta\cos^2\theta-\sin\theta}{a+1}=\frac{2\sin\theta\cos\theta}{a+2}\implies\frac1a=\frac{4\cos^2\theta-1}{a+1}=\frac{2\cos\theta}{a+2}\\\implies a^2-3a-4=0\implies a=4$$ by eliminating $\cos\theta$. Next, $\cos\theta=3/4$.

