Proof:
In ∆PMR,
QM = QR = a [Given]
∴ Q is the midpoint of side MR.
∴ seg PQ is the median.
∴ PM2 + PR2 = 2PQ2 + 2QM2 [Apollonius theorem]
∴ PM2 + a2 = 2a2 + 2a2
∴ PM2 + a2 = 4a2
∴ PM2 = 3a2
∴ PM, = (√3a) (i) [Taking square root of both sides]
SimlarIy, in ∆PNQ, R is the midpoint of side QN.
∴ seg PR is the median.
∴ PN2 + PQ2 = 2 PR2 + 2 RN2 [Apollonius theorem]
∴ PN2 + a2 = 2a2 + 2a2
∴ PN2 + a2 = 4a2
∴ PN = 3a2
∴ PN = (√3a) (ii) [Taking square root of both sides]
∴ PM = PN = √3a [From (i) and (ii)]