Given: E is the midpoint of HF and GK

Prove: HK = GF

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