$ABCD is a parallelogram with diagonal AC. If a line XY is drawn such that it cuts the side AD at point Z and XY || AB, then find the value of \frac{B X}{X C} .$

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Solution

$Consider Δ A B C,$

$A B ∥ X Y$

$∴ \frac{B X}{X C} = \frac{A Y}{Y C} . . . (i) (By Basic proportionality theorem)$

$Consider parallelogram ABCD, A B | | C D (Opposite sides of a parallelogram are parallel)$

$\Rightarrow A B ∥ C D ∥ X Z$

$Consider Δ A C D,$

$C D ∥ Y Z$

$∴ \frac{A Y}{Y C} = \frac{A Z}{Z D} . . . (i i) (By Basic proportionality theorem)$

$From (1) and (2),$
$\frac{B X}{X C} = \frac{A Y}{Y C} = \frac{A Z}{Z D}$

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