The
Circles (Lunes) & The Pythagorean Theorem
Problem
The
Circles (Lunes) & The Pythagorean Theorem
Problem
Given: In the figure above the blue triangle is a right
triangle with legs a and b and hypotenuse c.
Circles are drawn with diameters a, b, and c and with centers at the
midpoints of those sides of the right triangle.
Show: algebraically that the sum of the areas of the yellow
shapes (lunes) is equal to the blue area of the given right triangle.
Historical Note:
This problem is a generalization of the Theorem
of Hippocrates (ca. 430 B.C.), the mathematician,
not the physician.
That theorem, succinctly stated, is as follows:
Circles are to one another as the squares on their diameters.
This led to the following:
Similar segments of circles are in the same ratio as
the squares on their bases.
And this led to the problem of the quadrature of the lune.
A lune is a figure bounded by two circular arcs of unequal
radii.
All of this resulted as fallout in the pursuit of the famous,
ancient, and ultimately impossible "Squaring of the Circle" problem.
And as we now recognize, these figures are, respectively,
Pythagorean Similar.
The problem above then is to show that the sum of the areas
of the yellow lunes is equal to the area of the blue right
triangle.
Question:
Can you create the segment (and therefore the lune) on the
hypotenuse of the blue right triangle that is Pythagorean
Similar to the two other segments (black)?
Of course, the lune you create is Pythagorean Similar to
the two yellow lunes shown; and the sum of the areas of
the two yellow lunes will equal the area of your lune (which,
consequently, equals the area of the blue right triangle!).
