Spherical Pythagorean Theorem
Spherical Pythagorean Theorem
Define a "proper triangle" as a triangle having one angle equal to half the sum of its three interior angles. Call that angle the "proper angle," the side opposite to it the "hypotenuse," and the other two sides the "legs." In Euclidean geometry, the class of proper triangles corresponds to right triangles, for which the classical Pythagorean theorem holds. In spherical geometry, the theorem generalizes as follows: in proper spherical triangles (sides being arcs of great circles), the area of the circle having as radius the hypotenuse equals the sum of the areas of the circles having as radii the legs.