TITLE: Tangent circles in a rectangle
QUESTION [7 upvotes]: $ABCD$ is a rectangle with 4 semicircles tangent to each other. $AB$ is the diameter of a circle, $FC$, $GD$ are radii. Points $E$, $F$, $G$, $H$ are on the sides of the rectangle. The circle with radius $HI$ is tangent to the other circles. What is the ratio of $AB:BC$?

REPLY [2 votes]: Just to add the number crunching in M2. Jean Marie had two misprints in his equations (3) and (5). The solution point $(d,g,h,r,s)=(3,\frac{16}{25},\frac{45}{28},\frac{25}{28},\frac98)$ is found by the following few lines:
R=QQ[d,g,h,r,s] 
I=ideal(h^2+1-(r+1)^2,1+(d-s)^2-(1+s)^2,g^2+(d-h)^2-(r+g)^2,4+(d-s-h)^2-(r+s)^2,(2-g)^2+s^2-(s+g)^2) 
primaryDecomposition I -- {ideal(8*s-9,28*r-25,28*h-45,25*g-16,d-3), ideal(r+3*s+2,g*s+2*g-2,d*s-h*s-3*s^2-4*g-2*s+4,h^2-9*s^2-6*s,d*h-h*s-3*s^2,d^2-2*h*s-6*s^2-8*g-6*s+8)} 

So to answer the question, the ratio is $2:3$