Çözüldü Circles Tangent to One Another-Pythagorean Theorem-Quadratic Equation Factoring

For the high school classic exam:

What is the area in terms of r^2 of the smallest square bounded between and touched by two circles with radii r tangent to one another and to the same common tangent line, where the square's bottom side lies on this line and its other sides are constrained by the circles?

Side length of the square: x
2·[ r^2 - (r - x)^2 ]^0.5 + x = 2r ⇒ 5x^2 - 12r·x + 4r^2 = 0 ⇒ (5x - 2r)(x - 2r) = 0 ⇒ x = 2r / 5
Area(Square) = 4r^2 / 25.

Analysis and verification by the AI at www.mathful.com is attached, and its summary is below:

https://i72.servimg.com/u/f72/19/97/10/39/mathfu16.png

Inspired by the problem at: https: // www . youtube . com / watch?v=7RNJefEi_2s