000RM.dvi

540 The shoemaker’s knife

20.3 More Archimedean circles ..........

LetUVbe the external common tangent of the semicirclesO 1 (a)and
O 2 (b), which extends to a chordHKof the semicircleO(a+b). LetC 4
be the intersection ofO 1 VandO 2 U. Since

O 1 U=a, O 2 V =b, and O 1 P:PO 2 =a:b,

C 4 P=aab+b=t. This means that the circleC 4 (t)passes throughPand
touches the common tangentHKof the semicircles atN.

A O 1 O P O 2 B

N

U

V

H

K

C 4

M

C 5

LetM be the midpoint of the chordHK. SinceOandPare sym-
metric (isotomic conjugates) with respect toO 1 O 2 ,

OM+PN=O 1 U+O 2 V =a+b.

it follows that(a+b)−QM=PN=2t. From this, the circle tangent to
HKand the minor arcHKofO(a+b)has radiust. This circle touches
the minor arc at the pointQ.

Theorem 20.4(Thomas Schoch).The incircle of the curvilinear triangle
bounded by the semicircleO(a+b)and the circlesA(2a)andB(2b)has
radiust=aab+b.

A O 1 O P O 2 B

S

Proof.Denote this circle byS(x). Note thatSOis a median of the
triangleSO 1 O 2. By Apollonius theorem,

(2a+x)^2 +(2b+x)^2 =2[(a+b)^2 +(a+b−x)^2 ].

From this,x=aab+b=t.