Is triangle ABC a right-angled triangle, if BD=EC and angle ACB =60?

We are given that BD=DC and \(\angle ACB =60\)


(1) AB is parallel to DE.
AB || DE. Since DE makes a perpendicular on BC, AB will also make a perpendicular on BC.
Thus ABC is a right-angled triangle.

(2) D is the midpoint of AC.
We are not required to know related theorems in GMAT, so there has to be a simpler way.
Let us take \(\triangle BDC\) => \(BD=DC\), so \(\angle DBC = \angle DCB = 60\), so \(\angle BDC = 180-60-60=60\)
Let us now take \(\triangle BDA\) => \(\angle BDA = 180-\angle BDC =180-60=120\),\(BD=DA\), so \(\angle DBA = \angle DAB = \frac{60}{2}=30\), so \(\angle BDA=30\)
\(\angle ABC = \angle ABD+\angle DBC=60+30=90\)
Thus ABC is a right-angled triangle.

D