Using
circle inversion we can simplify things a lot. Instead of constructing
a chain of circles tangent to the semicircles, we can simply draw
circles between two lines. See below how it works!
First Tangent Circle
Drag point N
to
change the shape of the arbelos. Use the navigation buttons to go
through the construction steps for the first circle in the
chain.
We start with an arbelos and
a perpendicular line at its notch N. This gives us point P on the
large semicircle. Now, we use the green circle with center A
through point P to do a clever circle inversion:
The inversion of semicircle APB
is the blue line NP.
The inversion of semicircle AN is
the red line through
B parallel to NP.
With this circle inversion we have converted the
semicircles into lines. We can easily draw circle R'S'T'. This
circle is tangent to the two lines and the semicircle NB.
Now we go back and use a circle inversion at the green circle
again: The inversion of circle R'S'T' is circle RST which
is tangent to the semicircles. We have got our first circle in
the Pappus Chain!
Your Tasks
Explain why the semicircles become lines under the
circle inversions described in step 1.
Explain why the lines in step 1 go through N and B. To do
so, proof that point B is the result of a circle inversion of
point N at the green circle (i.e. show that AN ⋅ AB = AP2).
Hint: use the right triangle APB and its height NP.
Try to construct the first tangent circle on paper.
Five Circles of the Chain
To continue the chain, we can simply draw more circles between the two
dashed lines. Then we use circle inversion at the green circle to
convert them back into the arbelos.
Drag point N
to
change the shape of the arbelos in the construction below.