Dykstra Extension to the Pythagorean Theorem

Edsger W. Dykstra of the University of Texas at Austin proposes that the equation sgn(α + β - γ) = sgn(A2 + B2 - C2) is a superior, generalized form of the Pythagorean Equation.

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Using the Figure

Note that when H is concurrent with K, that gamma is π / 2, illustrating the limited case a2+b2=c2. However, due to limitations of digitized model, it is very difficult to make H exactly concurrent with K.

Construction

The figure is constructed as follows:
  1. Draw an arbitrary triangle ABC.
  2. Copy angle BAC onto BCK where K is the point where the ray defined by the angle intersects AB.
  3. Copy angle ABC onto ACH where H is the point where the ray defined by the angle intersects AB.

Proof

To prove: sgn(α + β - γ) = sgn(A2 + B2 - C2)

Take a triangle of sides A, B, and C, where the length of all the sides is greater than zero. Let the angles opposite each of the sides be α, β, and γ, respectively. A formal expression of the Pythagorean Theorem can be stated as:

γ = π / 2 → A2 + B2 = C2

The problem is with the transcendental π. Since α + β + γ = π, π can be eliminated. Elementary arithmetic gives:

γ = α + β → A2 + B2 = C2

Professor Dykstra proposes this can be strengthened to:

γ = α + β ≡ A2 + B2 = C2

While he does not propose a reason for this, I assume it is because Euclid has already proved the inverse of the Pythagorean Equation:

A2 + B2 = C2 ≡ γ = α + β

One can get an equivalent form by negating both sides:

A2 + B2 ≠ C2 ≡ γ ≠ α + β

Professor Dykstra then proposes, since the large angle is opposite the larger side, that:

A2 + B2 > C2 ≡ γ > α + β, and
A2 + B2 < C2 ≡ γ < α + β

These equivalencies can be jointly formulated using the sgn function, defined as sgn = ( sgn(x) = 0 ≡ x = 0 ) | ( sgn(x) = 1 ≡ x > 0 ) | ( sgn(x) = -1 ≡ x < 0). Note: a different formulation for the sign function, more amenable to computer math is sign(x) = x/abs(x) if x ≠ 0, otherwise sign(x)=0. This formulation is as follows:

sgn( α + β - γ ) = sgn( A2 + B2 - C2 )

QED

More Information

http://www.cs.utexas.edu/users/EWD/ewd09xx/EWD975.PDF