000RM.dvi

20.3 More Archimedean circles 541

Exercise

1.The circles(C 1 )and(C 1 ′)are each tangent to the outer semicircle

of the shoemaker’s knife, and toOQ 1 atQ 1 ; similarly for the circles

(C 2 )and(C 2 ′). Show that they have equal radiit=aab+b.

C 1

C 2

C 1 ′

C′ 2

A O 1 O P O 2 B

Q 1

Q 2

2.We call the semicircle with diameterO 1 O 2 themidwaysemicircle

of the shoemaker’s knife.

Show that the circle tangent to the linePQand with center at the

intersection of(O 1 )and the midway semicircle has radiust=aab+b.

C

C′

A O 1 O P O 2 B

Q

3.Show that the radius of the circle tangent to the midway semicircle,

the outer semicircle, and with center on the linePQhas radius

t=aab+b.

C

A O 1 O P O 2 B

Q