Given: Triangle ABC and ∆PQR in which AD and PM are medians drawn on sides BC and QR respectively. It is given thatTo Prove : ∆ABC ~ ∆PQRConst : Produce AD to E such that AD = DE and PM to F such that PM = MF.
Proof : In ∆ABD and ∆CDE,AD = DE    [by construction]∠ADB = ∠CDE[vertically opposite angles]and    BD = DC    [AD is a median]Therefore, by using SAS congruent condition                                   [by CPCT]Similarly, we can prove                                                [by CPCT]It is given that:              Therefore, by using SSS congruent condition                                                    ...(i)Similarly,                         ...(ii)Adding (i) and (ii), we get∠1 + ∠3 = ∠2 + ∠4∠A = ∠PNow, in ∆ABC and ∆PQR                     and           Therefore, by using SAS similar condition∆ABC ~ ∆PQR Hence Proved.