
Given that in ΔPQR, PQ = QR and, L M and N are mid-points of PQ, QR and RP respectively
We have to prove LN = MN
Join L and M, M and N, N and L
We have
PL = LQ, QM = MR and RN = NP
[ ∴ L, M and N are mid-points of PQ, QR and RP respectively]
And also
PQ = QR= PL = LQ = QM = MR=PQ/2=QR/2 ....(1) Using mid-point theorem, we have
MN || PQ and MN=(1/2) x PQ
⇒MN=PL=LQ .......(2)
Similarly, we have
LN QR and LN=(1/2) x QR
⇒ LM=QM MR .........(3)
From equation (1), (2) and (3), we have
PL= LQ= QM= MR= MN= LN
LN=MN