Given: E is the midpoint of HF and GK

Prove: HK = GF

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