Given: In Δ PQR, PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively
To prove: LN = MN
Join L and M, M and N, N and L
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 = PN = LR …….(i) [ Using mid-point theorem]
MN ∥ PQ and MN = PQ/2
MN = PL = LQ ……(ii)
Similarly, we have
LN ∥ QR and LN = (1/2)QR
LN = QM = MR ……(iii)
From equation (i), (ii) and (iii), we have
PL = LQ = QM = MR = MN = LN
This implies, LN = MN
Hence Proved.
