(Originais Teachy 2023) - Question Hard of Mathematics

An architect is designing a new building in a city where urban space is a scarce and expensive resource. To maximize land use, he decides that the shape of the building's base will be a circle. The architect wants the circle to have its center at point C(2, 5) and for a point A(4, 7) on the edge of the circle to also be one of the base points of the building. Additionally, the architect decides that the circle should pass through a point B(6, 1) which is located at a maximum distance of 6 units from the center of the circle. Considering this information, calculate the equation of the circle that will represent the building's base. Explain, in terms of analytic geometry, how the architect's decision to use a circle as the building's base can affect the design and structure of the building, considering also aspects of engineering and architectural design.