In a triangle ABC, median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram.
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 ΔABD and ΔXCD
⇒BD=DC, AD = DX and ∠BDA=∠CDX
Thus, By SAS, we say that ΔABD≅ΔXCD
Therefore, AB=XC. And also ∠ABD=∠XCD as alternate angles are equal. So, AB||XC.
Similar way, we can prove that the ΔDAC≅ΔDXB
So, the opposite sides are equals and also ∠ACD=∠XBD
We know that in a parallelogram, the diagonals bisect each other, opposite sides equal and parallel.
∴ ABXC is a parallelogram