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# If $ABCD$ is a parallelogram, $E$ and $F$ are the midpoint of $AB$ and $CD$ .Prove: $AF$ and $CE$ trisect the diagonal $BD.$

Last updated date: 15th Jul 2024
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Hint: In this problem, $ABCD$ is a parallelogram and the midpoint also given in the diagrammatic representation, Parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. Here, we use the conversion of the midpoint theorem, the line drawn through the midpoint of one side of a triangle, parallel to another side bisecting the third side.

In the given problem,
$ABCD$ is a parallelogram. $E$ and $F$ are the mid-points of sides $AB$ and $CD$ respectively.
To show: line segments $AF$ and $\;EC$ trisect the diagonal $BD$ .
Proof,
$ABCD$ is a parallelogram
Therefore, $AB\parallel CD$
also, $AE\parallel FC$
Now,
$AB{\text{ }} = {\text{ }}CD$
(Opposite sides of parallelogram $ABCD$ )
$\dfrac{1}{2}{\text{ }}AB = \dfrac{1}{2}CD$
$AE = FC$
Where, $E$ and $F$ are midpoints of side $\;AB$ and $CD$ .
Since a pair of opposite sides of a quadrilateral $\;AECF$ is equal and parallel.
so, $\;AECF$ is a parallelogram
Then, $AE\parallel EC$ ,
$AP\parallel EQ$ and $FP\parallel CQ$
Since, opposite sides of a parallelogram are parallel
Now,
In $\Delta DQC,$
$F$ is midpoint of side $DC$ and $FP\parallel CQ$
(as $AF\parallel EC$ ).
So, $P$ is the midpoint of $DQ$
Here, we use converse of midpoint theorem, we get
$DP = PQ{\text{ }} \to {\text{(1)}}$
Similarly,
In $APB,$
$E$ is the midpoint of side $AB$ and $EQ\parallel AP$ (as $AF\parallel EC$ ).
So, $Q$ is the midpoint of $PB$ .
By applying Converse of midpoint theorem, we get
$\;PQ = QB \to (2){\text{ }}$
From equations $(1)$ and $(2)$ , we get
$DP = PQ = BQ$
Hence, the line segments $AF$ and $EC$ trisect the diagonal $BD$ .

Note: Here, we have to solve the geometric problem by the diagrammatic representation and here, a parallelogram concept is used and a converse of the midpoint theorem is used. A quadrilateral is a parallelogram if Its opposite sides are equal, its opposite angles are equal, diagonals bisect each other and a pair of opposite sides is equal and parallel.