The given conditions and properties of midpoints, it is shown that AB = CD and AB = EF.
Here's a two-column proof for the given statement:
Statements Reasons
Given: G is the midpoint of AB, CD, and EF. Given
Given: AG = GE and GD = GB. Given
∵ G is the midpoint of AB → AG = GB. Definition of midpoint
∴ AG = GD. Transitive property of equality
∴ AG = GD = GB. Transitive property of equality
∵ G is the midpoint of CD → GD = GE. Definition of midpoint
∴ GD = GE. Substitution
∴ AG = GD = GB = GE. Transitive property of equality
∵ AG = GE → AB = EF (Using the transitive property). Definition of equality (midpoint of EF)
∵ AG = GD = GB → AB = CD (Using the transitive property). Definition of equality (midpoint of CD)
Hence, from the given conditions and properties of midpoints, it is shown that AB = CD and AB = EF.