UPSC 2017 CDS I Math Paper

In $\triangle ADC$ $\angle DAC = \angle DCA$ $\Rightarrow$ $CD = AD$Remember: Equal sides have equal angles opposite to them.Therefore, $\triangle ADC$ is an isosceles triangle.Hence, statement 1 is correct.In $\triangle ABC$,$AB = BC$ $\Rightarrow$ $\angle ACB = \angle BAC$$\therefore$ $\angle BAC - \angle DAC = \angle ACB - \angle DCA$$\Rightarrow$ $\angle BAD = \angle BCD$In $\triangle ABD$ and $\triangle CBD$,$AD = CD$ (Proved)$\angle BAD = \angle BCD$ (Proved)$AB = BC$ (Given)$\therefore$ $\triangle ABD \cong \triangle CBD$ (SAS congruence rule)Hence, statement 3 is correct.Now, D can be any point in the interior of the $\triangle ABC$ with $\angle DAC = \angle DCA$ or $CD = AD$So, $BD$ is not always equal to $AD$ and $DC$Hence, D is not the centroid of $\triangle ABC$.Hence, statement 2 is not correct.