Given: E is the midpoint of HF and GK

Prove: HK = GF

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