Let's call one triangle ABC, and M the midpoint of AB.
Let's call the other triangle DEF, and midpoint N of DE.
We have a congruent side, AB=DE, congruent medians CM=FN, and congruent angles, ∠AMC=∠DNF
AB ≅ DE Given
AM ≅ BM and DN ≅ EN Def median/midpoint
AM+BM=AB and DN+EN=DE Segment addition theorem
2AM=2BM=AB and 2DN=2EN=DE Substitution (eg BM for AM)
AM ≅ BM ≅ DN ≅ EN Transitivity, algebra
CM ≅ FN Given
∠AMC ≅ ∠DNF Given
ΔAMC ≅ ΔDNF Side Angle Side
AC ≅ DF CPCTC
∠BMC ≅ ∠ENF Supplements of congruent angles are congruent
ΔBMC ≅ ΔENF Side Angle Side
BC ≅ EF CPCTC
Δ ABC ≅ Δ DEF Side Side Side
