Notes on The Pythagorean Theorem
First consider Euclid's statement and proof of Proposition
In right-angled triangles the
square on the side opposite the right angle equals the sum of the squares
on the sides containing the right angle.
Note that Euclid's treatment in its statement or its "proof" never refers
the traditional equation, a2+b2=c2.
In one alternative proof for this theorem illustrated in the java sketch
below, we consider 4 congruent right triangles and 2 squares and then the
same 4 triangles and the square on the side of the hypotenuse arranged
inside of a square with side "a+b" . Can you explain how this sketch
justifies the theorem?
proof using "shearing" illustrated in the Java sketch below taken from
a Geometers' Sketchpad example can be connected to Euclid's proof.
(Based on Euclid's Proof) D. Bennett 10.9.9
Shear the squares on the legs by dragging point P, then point Q, to the
line. Shearing does not affect a polygon's area.
Shear the square on the hypotenuse by dragging point R to fill the right
The resulting shapes are congruent.
Therefore, the sum of the squares on the sides equals the square on the
In considering the Pythagorean theorem, what kind of assumptions were
needed in the first proof with the triangles and squares?
[Side Trip] Moving line segments:
Here are some considerations related to those assumptions:
How could we justify identifying "equal" objects (congruent figures)?
How do the objects fit together?
How do movements effect the shapes of objects?
Consider Euclid's Proposition
1 and Proposition
These propositions demonstrate that Euclid did not treat moving a line
segment as an essential property worthy of being at the foundations as
an axiom. However, this is a fundamental tool for all of geometry.
Note that in the proofs of propositions 1 and 2 certain points of
intersection of circles are presumed to exist without reference to any
of the postulates. These presumptions were left implicit for hundreds of
years, but were cleared up in the 19th century when careful attention was
given again to the axioms as a whole system. This presumption is sometimes
described as a postulate of "continuity" for lines and circles.
[An example of a geometry where circles do not
intersect is given by using the rational coordinate plane.
Points correspond to ordered pairs of rational numbers. then the circle
with center (0,0) and radius 1 and the circle with center (1,0) also radius
1 meet in the ordinary plane at the points with coordinates (1/2, sqrt(3)/2)
and (1/2, -sqrt(3)/2) . Since sqrt(3)/2 is not a rational number, this
ordered pair does not correspond to a point in the rational coordinate
plane, so the two circles do not have a point of intersection.
Another example of a point not in the rational
coordinate plane is the point (sqrt(2),0). This point can be constructed
in the ordinary plane with straight edge and compass using the circle with
center (0,0) and radius determined by the points (0,0) and (1,1). This
circle will meet the X-coordinate axis at the point (sqrt(2),0) ]