PQRS is a parallelogram. L and M are points on PQ and SR respecitvely such that PL = MR. Show that LM and QS bisect each other.

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Solution

Let LM and BAD intersect at O

as AL = CM and AB = CD
therefore AB-AL=CD-CM
BL = DM ___ (i)

consider $Δ$ OMD and $Δ$ OLB
$∠ D O M = ∠ B O L$ (vertically opposite angles)
$∠ O M D = ∠ O L B$ (alternate interior angles)
DM = BL (from i)
therefore $Δ$ OMD congruent to $Δ$ OLB
therefore OM = OL,

OD = OB (cpctc)

Therefore O bisects LM and BD
therefore LM, BD bisect each other

(here parellelogram is ABCD)

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