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Mathematical problems of nine circles
As shown in the figure:

According to the vertical diameter theorem:

R & ampsup2r & ampsup2= 3 & ampsup2=9

Semicircular area = π r&; sup2/2

Area of small semicircle = π r&; sup2/2

∴ Shadow area s = (π r&; sup2/2)-(πr & amp; sup2/2)=(π/2)(R & amp; sup2r & ampsup2)=9π/2