Two Semicircles and a Square in a Quarter Circle

This video is about inscribing two semicircles and a square in a quarter circle
If you need to post a picture of your solution or idea:
#GeometryPuzzles

[labzioui]

instead of « x+a » we must put :
«  x+ sqrt( 2ax-x^2) »

[madslarsen]

Thanks for all the nice geometry problems!
My solution for this one :
x = (2 * sqrt(5)/5) - (1- sqrt(5)/5)
x = (3 sqrt(5) - 5)/5

[labzioui]

With all due respect, in the 6 min 29 sec sequence of the video, "x + a" is false. Because the side of the square is not tangent to the circle of radius "a" at the point of intersection "square-circle ".