UPSC 2016 CDS I Math Paper

In $\triangle ABC$, $AB = AC$.Therefore,$\triangle ABC$ is an isosceles triangle.$L$ is the locus of point $X$ inside or on the triangle such that $BX = CX$. So, $X$ lies on the perpendicular bisector of the side $BC$. Now, the perpendicular bisector of an isosceles triangle is also the angle bisector of the triangle. So, incentre is the point of intersection of the angle bisectors of a triangle. Therefore, $L$ is a straight line passing through $A$, and incentre of triangle $\triangle ABC$ is on $L$.Remember:In an isosceles triangle, incentre, orthocentre, centroid and circumcentre lie on the same line So, $L$ is a straight line passing through $A$, orthocentre and centroid of triangle $\triangle ABC$. Hence, statements 1, 2 and 3 are correct.