Book XI · Proposition 20

XI.20

If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one.Heath, 1908

Every step, checked

• the solid angle is a genuine onegiven
• any two of the three plane angles, taken together, are greater than the remaining oneby XI.20

What it needs, and what needs it

Needs: nothing earlier.

Depth: 0 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition("XI.20", THEOREM, sample=samples3.tetrahedron)
def prop_XI_20(o: Point3, a: Point3, b: Point3, c: Point3) -> Out:
    """Any two plane angles of a solid angle exceed the third."""
    hypothesis("the solid angle is a genuine one", not coplanar(o, a, b, c),
               guard=True)
    for arm in (a, b, c):
        line3(o, arm)

    first, second, third = angle_at3(a, o, b), angle_at3(b, o, c), angle_at3(a, o, c)
    claim("any two of the three plane angles, taken together, are greater than "
          "the remaining one", "XI.20",
          all(_sum_exceeds(x, y, z) for x, y, z in
              ((first, second, third), (second, third, first), (first, third, second))))
    return Out(angles=(first, second, third))