Semi-Circles Within a Circle

[bob012345]

Here is a puzzle from Catriona Shearer. Find the area ratio of the shaded semi-circles to the circle which they fit in. The bases of the semi-circles are parallel.

Hint: consider the extreme cases to figure what the answer might be then prove the general case.

 

[anuttarasammyak]

NOT ELEGANT solution
The eqation of the circles
$$x^2+y^2=R^2$$
$$(x-x_1)^2+y^2=R^2-x_1^2$$
$$(x-x_2)^2+y^2=R^2-x_2^2$$
By deletion of the smaller two circles equatins, we see they touch at $x=x_1+x_2,y=0$ Thus
$$x_1^2=R^2-x_2^2$$
$$x_1^2+x_2^2=R^2$$
Area of the sum of small circles
$$2R^2-x_1^2-x_2^2=R^2$$
equals the area of the large circle. So the solution is 1/2.