000RM.dvi

536 The shoemaker’s knife

20.1 Archimedes’ twin circles ...........

LetPbe a point on a segmentAB. The region bounded by the three
semicircles (on the same side ofAB) with diametersAB,APandPBis
called a shoemaker’s knife. Suppose the smaller semicircles have radiia
andbrespectively. LetQbe the intersection of the largest semicircle with
the perpendicular throughP toAB. This perpendicular is an internal
common tangent of the smaller semicircles.

A O 1 O P O 2 B A O 1 OP O 2 B

Q

U

V

H

K

R

Theorem 20.1(Archimedes).The two circles each tangent toCP, the
largest semicircleABand one of the smaller semicircles have equal
radiit, given by

t=

ab

a+b

.

A O 1 O P O 2 B A O 1 O P O 2 B

Q