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In a triangle A B C median A D is produc...

In a triangle `A B C` median `A D` is produced to `X` such that `A D=D Xdot` Prove that `A B X C` is a parallelogram.

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In quadrilateral `ABXC`, we have
`AD=DX` [Given]
`BD=DC` [Given]
So, diagonals `AX` and `BC` bisect each other.
`therefore ABXC` is a parallelogram.
`/_BDA=/_CDX`
Now, we know that `/_\ABD~~/_\XCD`
Therefore, `XC=AB`
And we get that,
`/_ABC=/_XCB`
These two angles form a pair of alternate angles,
`therefore AB` parallel `CX`
Now, as two opposite sides are parallel and equal, we can say that `ABXC` is a parallelogram.
Hence Proved.
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