Find Area of square Geometry solution would be preferred?

You could describe everything in terms of functions

x^2 + y^2 = 1^2

(x - 1/2)^2 + y^2 = (1/2)^2

(x + 1/2)^2 + y^2 = (1/2)^2

E and F are going to have x-coordinates that sum to 0 (that is, one will be the negative of the other)  The distance between D and E will be equal to the distance between E and F.  That is, if E has an x-coordinate of -a/2, then F has an x-coordinate of a/2

Ex = -a/2

Ey : (-a/2 + 1/2)^2 + y^2 = 1/4

Ey : (1/4) * (1 - a)^2 + y^2 = 1/4

Ey : (1 - a)^2 + y^2 = 1

Ey : y^2 = 1 - (1 - a)^2

Ey : y^2 = 1 - (1 - 2a + a^2)

Ey : y^2 = 1 - 1 + 2a - a^2

Ey : y^2 = 2a - a^2

Ey : y = sqrt(2a - a^2)

x^2 + y^2 = 1

x = -a/2

y = Ey + a = a + sqrt(2a - a^2)

Now we can solve for a

(-a/2)^2 + (a + sqrt(2a - a^2))^2 = 1

(1/4) * a^2 + a^2 + 2 * a * sqrt(2a - a^2) + 2a - a^2 = 1

(1/4) * a^2 + a^2 - a^2 + 2a + 2a * sqrt(2a - a^2) = 1

(1/4) * a^2 + 2a + 2a * sqrt(2a - a^2) = 1

a^2 + 8a + 8a * sqrt(2a - a^2) = 4

a^2 + 8a - 4 = -8a * sqrt(2a - a^2)

(a^2 + 8a - 4)^2 = 64a^2 * (2a - a^2)

a^4 + 16a^3 - 8a^2 + 64a^2 - 64a + 16 = 128a^3 - 64a^4

a^4 + 64a^4 + 16a^3 - 128a^3 + 56a^2 - 64a + 16 = 0

65a^4 - 112a^3 + 56a^2 - 64a + 16 = 0

https://www.wolframalpha.com/input/?i=65a%5E4+-+11...

a = 0.28771 is the only answer that fits

a^2 is the area of the square

0.28771^2 = ‭0.0827770441‬