Geometric Problem. Find an angle. In△ABC , AB=AC ∠ACB=72∘. D is a point in △ABC such that ∠DBC=42∘,∠DCB=54∘.Find an angle∠BAD. Is there any geometric method? 
 A: Let, $AB=AC=a$ in isosceles $\triangle ABC$ 
Using Sine rule  in isosceles $\triangle ABC$  $$\frac{\sin \angle BAC}{BC}=\frac{\sin \angle ABC}{AC}$$ $$\frac{\sin 36^\circ}{BC}=\frac{\sin 72^\circ}{a}\implies BC=2a\sin 18^\circ$$ Similarly applying Sine rule in $\triangle BCD$
$$\frac{\sin \angle DCB}{BD}=\frac{\sin \angle BDC}{BC}$$  $$\frac{\sin 54^\circ}{BD}=\frac{\sin 84^\circ}{2a\sin 18^\circ}$$  $$\implies BD=\frac{2a\sin 18^\circ\sin 54^\circ}{\cos 6^\circ}$$
Let, $\angle BAD=\alpha$
Applying Sine rule in $\triangle ABD$
$$\frac{\sin \angle BAD}{BD}=\frac{\sin \angle ADB}{AB}$$  $$\frac{\sin \alpha}{\frac{2a\sin 18^\circ\sin 54^\circ}{\cos 6^\circ}}=\frac{\sin (180^\circ-(\alpha+30^\circ))}{a}$$ $$\cos 6^\circ\sin \alpha=2\sin 18^\circ\sin 54^\circ\sin (\alpha+30^\circ)$$  $$\frac{\sin(\alpha+30^\circ)}{\sin \alpha}=\frac{\cos 6^\circ}{2\sin 18^\circ\sin 54^\circ}$$ $$\frac{\sin \alpha\cos 30^\circ+\cos \alpha\sin 30^\circ}{\sin \alpha}=\frac{\cos 6^\circ}{2\sin 18^\circ\sin 54^\circ}$$
$$\frac{\sqrt 3}{2}+\frac{1}{2}\cot \alpha=\frac{\cos 6^\circ}{2\sin 18^\circ\sin 54^\circ}$$
$$\frac{1}{2}\cot \alpha=\frac{\cos 6^\circ}{2\sin 18^\circ\sin 54^\circ}-\frac{\sqrt 3}{2}$$
$$\cot \alpha=\frac{\cos 6^\circ-\sqrt 3\sin 18^\circ\sin 54^\circ}{\sin 18^\circ\sin 54^\circ}$$
$$\tan \alpha=\frac{\sin 18^\circ\sin 54^\circ}{\cos 6^\circ-\sqrt 3\sin 18^\circ\sin 54^\circ}$$
$$\bbox[5px, border:2px solid #C0A000]{\color{red}{\alpha=\tan^{-1}\left(\frac{\sin 18^\circ\sin 54^\circ}{\cos 6^\circ-\sqrt 3\sin 18^\circ\sin 54^\circ}\right)=24^\circ}}$$
A: This problem is contradictory and has no solution. Here are the details: Let x=angle BAd, y=angle DAC. Extend DA to E and let EDC=t,EDB=z. Then we have z=x+30,t=y+18,t+z=42,x+y=36. We get the following matrix 
1   0   -1  0   -30
0   1   0   -1  -18
0   0   1   1   42
1   1   0   0   36
                If we use Gauss elimination we get the last row being 
0 0 0 0 42
This means 0=42 which is absurd.
