A Shoemaker's Knife
An arbelos refers to the area defined by two semicircles inscribed in a third, larger semicircle. The shape resembles the blade of a knife used by Greek shoemakers, called "arbelos."
Archemides and Arbelos
It is believed that Archemides was one of the first mathematicians to have studied the properties of arbelos. The two congruent circles inscribed in an arbelos bear his name.
A series of inscribed circles, with any two adjacent ones tangent to each other, form a Pappus chain. Pappus of Alexandria was a Greek mathematician who studied the properties of arbelos.
Three different semicircles describe two different paths. Which one is shorter, the blue one or the yellow one?
Area of the Arbelos
The segment is tangent to the two smaller semicircles. Show that the area of the arbelos equals the area of the circle whose diameter is the given segment.
The segment is tangent to the two smaller semicircles, and the two inscribed circles are tangent to everything they touch. Show that the circles have the same radius.
The First Pappus Circle
The circle of radius r is tangent to each semicircle. Show that its center is 2r distance above the horizontal segment.
The Bankoff Circle
Take the two smaller semicircles. Their own tangency point and their tangency points to the first Pappus Circle form a new circle. Show that it is congruent
to the Archemides' Circles.