Given: E is the midpoint of HF and GK

Prove: HK = GF

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