Given that, in ΔPQR, PQ =QR and L,M,N are midpoints of the sides PQ, QP and RP respectively and given to prove that LN = MN

Here we can observe that PQR is and isosceles triangle

And also, L and M are midpoints of PQ and QR respectively

∴ ∠QPR and ∠LPN and ∠QRP and ∠MR are same

PN=NR [∴ N is midpoint of PR]

So, by SAS congruence criterion, we have ΔLPN≅ΔMRN

⇒ LN=MN

[∴ Corresponding parts of congruent triangles are equal]