Question 1198960
<br>
Here is a diagram:<br>
{{{drawing(400,400,0,800,0,800
,line(200,200,500,200),line(200,200,200,500),line(500,200,500,500),line(200,500,500,500)
,circle(200,200,150*sqrt(2)),circle(500,500,150*sqrt(2))
,circle(500,200,300-150*sqrt(2)),circle(200,500,300-150*sqrt(2))
)}}}<br>
Let {{{2x}}} be the side length of the square.  Then...
the diagonal of the square is {{{2x*sqrt(2)}}},
so the radius of the larger circles is {{{x*sqrt(2)}}},
so the radius of the smaller circles is {{{2x-x*sqrt(2)=x(2-sqrt(2))}}}<br>
The problem asks for the factor by which the radius of the smaller circles has to be multiplied to get the radius of the larger circles.  That factor is<br>
{{{(x*(sqrt(2)))/(x*(2-sqrt(2)))=(sqrt(2))/(2-sqrt(2))=((sqrt(2))(2+sqrt(2)))/((2-sqrt(2))(2+sqrt(2)))=(2*sqrt(2)+2)/(4-2)=(2*sqrt(2)+2)/2=sqrt(2)+1}}}<br>
ANSWER: {{{sqrt(2)+1}}}<br>