000RM.dvi

538 The shoemaker’s knife

20.2.2 Bankoff’s constructions

Theorem 20.3(Leon Bankoff).If the incircleC(ρ)of the shoemaker’s
knife touches the smaller semicircles atXandY, then the circle through
the pointsP,X,Yhas the same radius as the Archimedean circles.

A O 1 O P O 2 B

C

Z

X

Y

Proof.The circle throughP,X,Yis clearly the incircle of the triangle
CO 1 O 2 , since

CX=CY=ρ, O 1 X=O 1 P=a, O 2 Y =O 2 P=b.

The semiperimeter of the triangleCO 1 O 2 is

a+b+ρ=(a+b)+

ab(a+b)

a^2 +ab+b^2

=

(a+b)^3

a^2 +ab+b^2

.

The inradius of the triangle is given by
√
abρ
a+b+ρ

=

√

ab·ab(a+b)

(a+b)^3

=

ab

a+b

.

This is the same ast, the common radius of Archimedes’ twin circles.

First construction

A O 1 O P O 2 B

C

X

C 3 Y

Q 1

Q 2

Z