In a parallelogram ABCD, points M and N have been taken on opposite sides AB and CD respectively such that AM = CN.
Question:

In a parallelogram ABCD, points M and N have been taken on opposite sides AB and CD respectively such that AM CN. Show that AC and MN bisect each other.

Solution:

Given: In a parallelogram ABCD, AM CN.
 
To prove: AC and MN bisect each other.
 
Construction: Join AN and MC.

Proof:
Since, ABCD is a parallelogram.

$\Rightarrow A B \| D C$

$\Rightarrow A M \| N C$

Also, AM = CN           (Given)

Thus, AMCN is a parallelogram.

Since, diagonals of a parallelogram bisect each other.

Hence, AC and MN bisect each other.

 

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