Construction of a square with area the same
as that of a rectangle with sides a and b.
Given two segments of lengths a and b, to construct
the segment with length equal to the square root of a.b, we proceed
as follows:
-
Line up the two segments of length a and b on a single
line.
-
Draw the circle of center in the midpoint of the lined up
segments.
-
At the common extremity D to the two segments draw a perpendicular
segment to the diameter, meeting the circle at the point C.
CLAIM: The
square on the segment CD has the same area as the rectangle with sides
of length a.b
WHY?
-
Any triangle inscribed in a semi-circle
is a right triangle!
-
Triangles ABC, ACD and CBD are all similar right triangles.
-
Using the smaller triangles we have
the ratios between the corresponding
sides are the same
(being the magnification factor relating the two triangles):
-
so AD/CD
= CD/BD,
-
or by "cross multiplying"
CD.CD
= AD.BD
Conclusion: The
square on the side CD has the same area as the rectangle with sides a and
b.