Bisect $AM$ at $N$. Construct equilateral $\triangle$s $NME, MNO$. Then $AN=NM=ME=EN=NO=OM=BM$.
Thus $\triangle$s $EOM, OEN$ have a common base $EO$ and all four short sides are equal, so they are congruent (SSS); their obtuse angles are $120^\circ$, so their acute angles are $30^\circ$.
Each of the angles $OMB, BME, ENA, ANO$ is an exterior angle of an equilateral $\triangle$ and is thus $120^\circ$, so the $\triangle$s $OBM, BEM, EAN, AON$ are congruent to the above $\triangle$s $EOM, OEN$ (SSA).
Thus $AE=AO=EO=BE=BO$ so $\triangle$s $AOE, BEO$ are equilateral.
Thus $O$ is the centre of the circle through $A, E, B$. $\angle AEB=120^\circ=\angle ACB$, so $C$ lies on this same circle.
$BMN$ is a straight line and $\angle BMC=60^\circ$, so $\angle CMN=120^\circ$; $\triangle MNO$ is equilateral, so $\angle NMO=60^\circ$, so $OMC$ is also a straight line.
$\angle NOA=30^\circ$, so $\angle COA=90^\circ$. Thus $\angle CBA=45^\circ$ as the angle at the circumference is half the angle at the centre.