Problem
In $\triangle{ABC}$, $\angle ABC=45^ \circ$. $X$ is a point on $BC$ such that $BX=\frac{1}{3}BC$ and $\angle AXC=60^ \circ$. Find $\angle ACB$.
The problem looks easy. Though I couldn't solve it in an efficient way. Finally I solved it using trigonometry.
Trig solution
Let $BX = a$ units, then $BC = 3a$ and $XC = 3a-a= 2a$ units. $\angle AXC =60^ \circ$ and $\angle ABC= 45^ \circ$, then $\angle BAX= 60^ \circ -45^ \circ = 15^ \circ$.
Applying sine rule in $\triangle ABX$,
$$\frac{BX}{\sin \angle BAX} = \frac{AX}{\sin \angle ABC}$$
$$\implies \frac{a}{\sin 15°} = \frac{AX}{\sin 45°} \tag{1}$$
In $∆AXC$ , let $\angle ACB = \theta$, then $\angle XAC = (120 – \theta)$ and by sine rule,
$$\frac{XC}{\sin \angle XAC} = \frac{AX}{\sin \angle ACB}$$
$$\implies \frac{2a}{\sin (120 – \theta)} = \frac{AX}{\sin \theta} \tag{2}$$
Dividing $(1)$ by $(2)$,
$$\frac{\sin (120-\theta)}{2\sin 15^ \circ}= \frac{\sin \theta}{\sin 45°}$$
$$\implies 2\sin 15°\cdot\sin \theta = \sin 45°\cdot\sin (120-\theta)$$
$$\implies \frac{\sqrt{3}–1}{\sqrt 2}.\sin \theta = \frac{1}{\sqrt 2}.(\sin120°.\cos \theta – \cos 120°.\sin \theta).$$
$$\implies (\sqrt{3}–1).\sin \theta = \frac{\sqrt 3}{2}.\cos \theta + \frac{1}{2}.\sin \theta$$
$$\implies \tan \theta= 2+\sqrt 2$$
$$\implies \theta=75^ \circ$$
Thus, $\angle ACB = 75°$.
This solution is impossible without knowing the values of $\sin 15^ \circ$ and $\tan 75^ \circ$. And I find trigonometry boring. So, can this problem be solved in some other ways?
Best Answer
Drop a perpendicular from point $C$ on $AX$ and let the feet of this perpendicular be $D$.
$DX=\frac {1}{2}XC=BX$ and thereafter $BD=DC$ and $AD=BD$ by simple angle chasing. Thus $D$ is the circumcentre of $\triangle ABC$ and $\angle ACB=90-15=75^{\circ}$