The diagonal SQ of the parallelogram PQRS is divided into three equal parts at the points K and L. PK intersects SR at the point M and RL intersects PQ at the point N. Let us prove that, PMRN is a parallelogram.

Figure according to given information

SK = KL = QL … given … (i)

Consider ΔPKQ and ΔRLS

PQ = SR … opposite sides of parallelogram PQRS

∠PQK = ∠RSL … alternate pair of interior angles for parallel lines PQ and SR with transversal as SQ

SL = SK + KL and KQ = KL + LQ so using (i) we can say that

SL = KQ

Therefore, ΔPQK ≅ ΔRSL

⇒ ∠PKQ = ∠RLS … corresponding angles of congruent triangles

Thus PM || NR because ∠PKQ and ∠RLS are pair of alternate interior angles with transversal as KL

⇒ PM || NR … (ii)

Consider ΔPMS and ΔRNQ

∠PMS = ∠NRM … corresponding pair of angles for parallel lines PM and NR with transversal as SR … (a)

∠NRM = ∠RNQ … alternate interior angles for parallel lines PQ and SR with transversal as NR … (b)

⇒ ∠PMS = ∠RNQ … using equation (a) and (b)

∠PSM = ∠RQN … opposite pair of angles for parallelogram PQRS

PS = QR … opposite pair of sides for parallelogram PQRS

Therefore, ΔPMS ≅ ΔRNQ … AAS test for congruency

⇒ PM = NR … corresponding sides of congruent triangles … (iii)

⇒ SM = NQ … corresponding sides of congruent triangles … (c)

As PQ = SR … opposite sides of parallelogram PQRS … (d)

From figure PN = PQ – NQ and MR = SR – SM

Using (c) and (d)

PN = SR – SM

⇒ PN = MR … (iv)

As PQ || SR and PN and MR lie on the lines PQ and SR respectively hence we can conclude that

PN || MR … (v)

Using equations (ii), (iii), (iv) and (v) we can conclude that for quadrilateral PMRN the opposite sides are congruent and parallel therefore, PMRN is a parallelogram