A well-known problem that dates back to antiquity.
Find the shaded area in terms of
a
If the radii of the small circles are
r sub one
r sub two
cap r
cap r equals r sub one plus r sub two
By the Intersecting Chords Theorem (Euclid Bk 2 Prop 35) we have
two times r sub one multiplication two times r sub two equals a squared
four times r sub one times r sub two equals a squared
The shaded area is
pi times cap r squared minus pi times r sub one squared minus pi times r sub two squared equals pi times left parenthesis r sub one plus r sub two right parenthesis squared minus pi times r sub one squared minus pi times r sub two squared
and
sum with 3 summands pi times r sub one squared plus pi times r sub one squared plus two times pi times r sub one times r sub two minus pi times r sub one squared minus pi times r sub two squared equals two times pi times r sub one times r sub two
which equals
pi times a squared solidus two
