Given that,

In Δ PQR

PQ = QR

And,

L, M, and N are the mid points of PQ, QR and RP respectively

To prove: LM = MN

Construction: Join L and M, M and N and N and L

Proof: We have,

PL = LQ, QM = MR and RN = NP

Since, L, M and N are mid points of PQ, QR and RP respectively.

And, also PQ = QR

PL = LQ = QM = MR = = (i)

Using mid-point theorem, we have

MN ‖ PQ

And,

MN = PQ = MN = PL = LQ (ii)

Similarly, we have

LN ‖ QR

And,

LN = QR = LN = QM = MR (iii)

From equations (i), (ii) and (ii), we have

PL = LQ = QM = MR = MN = LN

Therefore, LN = MN

Hence, proved