Question

In a triangle ABC, median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram.

Answer 1

​Since AD is the median, it bisects the side BC.
Hence, BD = DC.

Also given that X is defined such that AD = DX.
Hence, the diagonals AX and BC of a quadrilateral ABXC bisect each other.
And also consider triangles $\Delta ABD$ and $\Delta XCD$
$\Rightarrow BD=DC$, AD = DX and $\angle BDA=\angle CDX$
Thus, By SAS, we say that $\Delta ABD\cong \Delta XCD$
Therefore, $AB=XC$. And also $\angle ABD=\angle XCD$ as alternate angles are equal. So, $AB||XC$.
Similar way, we can prove that the $\Delta DAC\cong \Delta DXB$
$\Rightarrow AC=BX$
So, the opposite sides are equals and also $\angle ACD=\angle XBD$
AC||XB

We know that in a parallelogram, the diagonals bisect each other, opposite sides equal and parallel.
$\therefore$ ABXC is a parallelogram