In order to construct the Archimedian Twin Circles we need to know
their radius. Right now, we don't know yet if they really both
have the same radius, so let's find out!
In the sketch below you
see an arbelos with one of the twin circles. You can drag point N along
the diameter to change the shape of the arbelos.
Your task is to calculate the twin circle's radius r in terms
of the two small semicircles' radii a and b. Note: the large
semicircle has center point M
and its radius is R = a
+ b.
Use the Pythagorean theorem for the right triangles CEF and
MEF. This lets you express the segment EF in two ways. Hint: Express
EF2
rather than EF to avoid square
roots.
Use the two equations from (1) to get one equation without
segment EF. Hint: Your result
should only contain the segments CE, CF, ME and MF.
Let's now try to get a formula for r in terms of a and b.
Start with writing all segments used in (2) in terms of a, b, r and R. Hint
1: CE,
CF and ME can be expressed directly. Hint 2:
MF
= MC - CF where MC = MA - AC Hint 3:
Your
results for the segments should be a - r,
R - r, R - 2a + r
and a + r.
Now plug your results from (3) into the equation from (2)
and solve it to get our twin circle's radius r. Hint
1: Use R = a + b to get a
result for r
that only depends on a
and b. Hint 2:
Your solution should be r = ab / (a + b).
Right
Twin Circle
Now you know how to get the radius r
of the left twin circle. Does the right
twin circle have the same radius? Let's make sure and proof
it! The construction below shows both twin circles.
Calculate the radius r
of the right twin circle by using the same strategy as above.
Do you
get the same result?
Archimedes'
Proof
You would like to know how Archimedes proved that the twin circles have same size? Find out in Proposition 5 of his Book of Lemmas.