Book VI · Proposition 24

VI.24

In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.Heath, 1908

Every step, checked

• the parallelogram is genuinegiven
• the division is a proper onegiven
• the lesser parallelogram stands about the same diameterby I.43
• and it is similar to the wholeby VI.Def.1

What it needs, and what needs it

Needs: I.43

Used by: VI.26

Rests on: C.N.1, C.N.2, C.N.3, C.N.4, C.N.5, Def.10, Def.15, Def.4, Post.5, VI.Def.1

Depth: 12 steps of argument above the first principles. Parallel postulate: needed.

What it takes on trust

• a circle-circle intersection is used, though no postulate asserts that two circles meet (×2)continuity
• a line-circle intersection is used, though no postulate asserts that a line meets a circle (×2)continuity

The proposition as code

@proposition(
    "VI.24",
    THEOREM,
    sample=_parallelogram_with_similar_corner,
)
def prop_VI_24(a: Point, b: Point, d: Point, part) -> Out:
    """The parallelograms about the diameter of ABCD."""
    hypothesis("the parallelogram is genuine", not collinear(a, b, d), guard=True)
    hypothesis("the division is a proper one", sign(part) > 0 and sign(1 - part) > 0, guard=True)
    whole = _parallelogram_on(a, b, d)
    outline(*whole)
    diameter = line(a, whole[2], "the diameter AC")

    corner = posit(_along(a, whole[2], part), "K")
    inner = _parallelogram_on(a, posit(_along(a, b, part), "E"),
                              posit(_along(a, d, part), "G"))
    outline(*inner)

    because(prop_I_43, whole[0], whole[1], whole[2], whole[3], corner)

    claim("the lesser parallelogram stands about the same diameter", "I.43",
          on_line(corner, diameter) and inner[2] == corner)
    claim("and it is similar to the whole", "VI.Def.1",
          similar((inner[0], inner[1], inner[2]), (whole[0], whole[1], whole[2])))
    return Out(inner=inner, whole=whole)