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

To Prove: ABXC is a parallelogram

Given: AD is a median of triangle ABC, produced to X and AD = DX

Concept Used:

SAS Theorem: If two sides and one angle of a triangle are equal to two sides and one angle of another triangle, the triangles are congruent.

In two congruent triangle, all angles and all sides are equal.

Diagram:

Proof:

In ΔABD and ΔXCD

AD = XD

BD = CD

∠BDA = ∠CDX

Now, we know that ΔABD ≈ ΔXCD

Therefore,

XC = AB

And we get that,

∠ABC = ∠XCB

These two angles form a pair of alternate angles, therefore,

AB || CX

Now, as two opposite sides are parallel and equal, we can say that ABXC is a parallelogram.