The ARBELOS

Leon Bankoff's "Surprises"
12 More Arbelos Twins, by Thomas Schoch.

The following interactive applet is done by me, and we teach you how to make applets using the language Java.

If you don't see a picture, perhaps you are not using a browser such as Netscape 3.0 running on Windows95 that can run Java applets. Sorry.

Notice the 4 white and 5 red circles, called Archimedean clones, always have the same size. Each of them obeys different constraints. The part bounded by the 3 yellow semicircles is called the Arbelos. It has the same area as the big blue circle. The small blue circle touches the 3 yellow arcs. Two of the quintuplets each touch 2 of the yellow arcs plus the vertical line at D. The blue vertical line goes through the centers of the red circles. To learn the geometric proofs you have to learn circle inversions. This is part of the modern geometry we teach at Biola, and is one of the most beautiful part of geometry that God creates and man discovers.

This chapter in geometry exemplifies many interesting things which we teach math students at Biola.

AC is a fixed horizontal line segment, and C1 a semicircle on AC as diameter. If you run our computer program "arbelos", you can use the mouse to move the point B on AC. Then the program will redraw the semicircles C2 and C3. The portion ouside C2, C3 and inside C1 is called the arbelos, or shoemaker's knife, in Greek. It is studied by Archimedes of old, and recently by Victor Thebault in France and the dentist Leon Bankoff of Beverly Hills. The following facts ("theorems") are stunningly beautiful. It should entice you to come to Biola.

Thm 1: Let MN be exterior common tangent to C2 and C3, and BD the vertical interior common tangent. Then MN = BD, and BMDN is a rectangle inscribed in a circle C9 that has the same area as the arbelos itself. Wow!

Thm 2: Let C5 be a circle touching C1, C2, and BD. Let C6 similarly touch C1, C3, and BD. Then, ysteriously, they have the same radii, regardless of how you move the point B on line AC. So we call them the twins. Let C8 be the largest circle touching C1 and the line MN, lying in the minor segment that line MN cuts off from C1. Then C8 has the same radius as C5 and C6. So they are triplets.

Thm 3: Let C4 be the circle touching C1, C2, C3, at points R, P, Q, inside the arbelos. Let C7 be the circle through B, P, Q. Then the center of C7 lies on BD, and C7 has the same radius as C5, C6, C8. So we have the quadruplets! (See diagram at Thm 6.)

Thm 4: Let C10 be the circle with center on BD, and going through B and R. Then it touches C1 at R. Let a, b be the diameters of C2 and C3. Then the diameter of C10 is the harmonic mean H.M. of a,b. The diameter BD of C9 is the geometric mean G.M. of a,b. The height of the arbelos is the arithmetic mean A.M. of a,b. So obviously the diagram demonstrates that H.M. < G.M. < A.M.

Thm 5: The points R,P,B,C lie on some circle C12 whose center is on the mid point of arc BC of C3. The points R,Q,B,A lie on some circle C11 whose center is on the midpoint of arc AB of C2.

Thm 6: The center of C4 is twice as far from AB as the lowest point on C4 is from AB.

The proofs of these theorems run into more than 6 handwritten pages of notes, unless we teach you the basic techniques of geometric circular inversions. With such tools you can prove all the above theorems in just 2 pages. Victor Thebault is a wizard with inversions, I was told. He died, leaving 5 chapters of an unpublished book on the arbelos to the dentist Bankoff, who is 84, trying to write 5 more chapters, and claiming to be the person on earth most knowledgeable about the arbelos today.