Book VI · Proposition 26

VI.26

If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, it is about the same diameter with the whole.Heath, 1908

The source used here misprints this enunciation — it reads “diameter with the whole” where Heath has “diameter with the whole.” (the closing full stop is missing). The full errata →

The converse of VI.24, and the step VI.28 needs to know where the deficient parallelogram sits.

Every step, checked

• the parallelogram is genuinegiven
• the division is a proper onegiven
• the lesser is similar to the whole and shares the angle at Agiven
• the far corner of the lesser lies on the diameter of the wholeby VI.24

What it needs, and what needs it

Needs: VI.24

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

What it takes on trust

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

The proposition as code

@proposition(
    "VI.26",
    THEOREM,
    sample=_parallelogram_with_similar_corner,
    note="The converse of VI.24, and the step VI.28 needs to know where the "
    "deficient parallelogram sits.",
)
def prop_VI_26(a: Point, b: Point, d: Point, part) -> Out:
    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)
    inner = _parallelogram_on(a, posit(_along(a, b, part), "E"),
                              posit(_along(a, d, part), "G"))
    outline(*inner)
    hypothesis("the lesser is similar to the whole and shares the angle at A",
               similar((inner[0], inner[1], inner[2]), (whole[0], whole[1], whole[2])))

    diameter = line(a, whole[2], "the diameter AC")
    because(prop_VI_24, a, whole[1], whole[3], part)

    claim("the far corner of the lesser lies on the diameter of the whole", "VI.24",
          on_line(inner[2], diameter))
    return Out()