EMAT 6700
Construct a rectangle ABED. Construct a circle with
radius DE and center D. Construct a segment DG equal to the radius
of DE extending segment AD. Find the midpoint of this new segment
AG, point H. Draw a semicircle with center at H and radius of AH.
Draw a perpendicular line from segment AD at point D until it intersects
with the semicircle at point J. Draw a circle with center J and radius
JD. Draw another radius from point J perpendicular to DJ, labeled
JM. Finish the square JDMN. The area of rectangle ABDE is equal
to the square DJMN. (see below for proof)(for an animation click here)
Proof: AH=DJ=DG since they are all radii of the semicircle. Triangle HDJ is a right triangle by construction.
Let HJ = a, let HD = b, and let DJ = c. Area
of Rectangle ABED = base times the height
= AD x ED
= AD x GD (ED=GD, radii are equal)
= (a+b)(a-b)
= a² - b²
= c² = Area of JDMN
(Dunham)