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Four generalizations of the Pythagorean theorem

The Endeavour 2025-11-13

Here are four theorems that generalize the Pythagorean theorem. Follow the links for more details regarding each equation.

1. Theorem by Apollonius for general triangles.

a^2 + b^2 = 2(m^2 + h^2)

2. Edsgar Dijkstra’s extension of the Pythagorean theorem for general triangles.

\text{sgn}(\alpha + \beta - \gamma) = \text{sgn}(a^2 + b^2 - c^2)

3. A generalization of the Pythagorean theorem to tetrahedra.

V_0^2 = \sum_{i=1}^n V_i^2

4. A unified Pythagorean theorem that covers spherical, plane, and hyperbolic geometry.

A(c) = A(a) + A(b) - \kappa \frac{A(a) \, A(b)}{2\pi}

The post Four generalizations of the Pythagorean theorem first appeared on John D. Cook.