Book VI · Proposition 23

VI.23

Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.Heath, 1908

'Compounded ratio' is the product of two ratios, which Euclid has no notation for and states by naming a mean.

Every step, checked

• neither parallelogram is degenerategiven
• they are equiangulargiven
• the ratio of the parallelograms is that of the sides compoundedby VI.1, V.Def.5

What it needs, and what needs it

Needs: VI.1

Used by: VI.27

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, V.Def.5

Depth: 15 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.23",
    THEOREM,
    sample=lambda rng: _two_parallelograms(rng, equal=False),
    note="'Compounded ratio' is the product of two ratios, which Euclid has no "
    "notation for and states by naming a mean.",
)
def prop_VI_23(a: Point, b: Point, d: Point, p: Point, q: Point, s: Point) -> Out:
    hypothesis("neither parallelogram is degenerate",
               not collinear(a, b, d) and not collinear(p, q, s))
    hypothesis("they are equiangular", eq_angle(b, a, d, q, p, s))
    first = _parallelogram_on(a, b, d)
    second = _parallelogram_on(p, q, s)
    outline(*first)
    outline(*second)

    # VI.1 compares figures on one straight line, and these parallelograms share
    # no such line until one is carried over to meet the other. See VI.14.

    claim("the ratio of the parallelograms is that of the sides compounded",
          ["VI.1", "V.Def.5"],
          _area(*first) * (length(p, q) * length(p, s))
          == _area(*second) * (length(a, b) * length(a, d)))
    return Out()