Given: E is the midpoint of HF and GK

Prove: HK = GF

CPCTC
Definition
HK = GF
∠HEK = ∠FEG
E is the midpoint of HF and GK
△HEK = △FEG
HE = FE and KE = GE
Given
SAS (Side-Angle-Side)
Definition
∠HEK and ∠FEG are vertical angles
Vertical angles are equal