One of the Ancient Greeks famous problem entails, the construction of square whose area equals that of circler using only compass and straightedge. Research this problem and discuss it? Is it possible? If no or yes, explain providing clear rational?

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The problem of constructing a square with the same area as a given circle using only a compass and straightedge is known as "squaring the circle." The ancient Greeks were indeed intrigued by this problem, as it presented a challenge in geometric constructions.

However, it was proven to be impossible to solve using only compass and straightedge in the 19th century. This impossibility is known as the "impossibility of squaring the circle" or "Lindemann–Weierstrass theorem." The theorem states that it is impossible to construct, using only compass and straightedge, a square with an area exactly equal to the area of a given circle.

The proof of this theorem relies on the fact that the mathematical constant pi (π) is transcendental, meaning it is not the root of any non-zero polynomial equation with rational coefficients. The circle's area is proportional to π, and because π is transcendental, it cannot be constructed using only compass and straightedge, which can only produce algebraic numbers.

Therefore, the problem of squaring the circle cannot be solved with the restrictions of compass and straightedge constructions.