Book I · Proposition 43

I.43

In any parallelogram the complements of the parallelograms about the diameter are equal to one another.Heath, 1908

Every step, checked

• ABCD is a parallelogramgiven
• K lies on the diameter AC, strictly insidegiven
• the diameter bisects the whole parallelogramby I.34
• and it bisects each of the parallelograms about itby I.34
• so the complements EBGK and HKFD are equalby C.N.3

What it needs, and what needs it

Needs: I.31 I.34

Used by: I.44 II.4 II.5 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

Depth: 11 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(
    "I.43",
    THEOREM,
    sample=_parallelogram_with_diameter_point,
)
def prop_I_43(a: Point, b: Point, c: Point, d: Point, k: Point) -> Out:
    hypothesis("ABCD is a parallelogram",
               parallel(Line.through(a, b), Line.through(d, c))
               and parallel(Line.through(a, d), Line.through(b, c)))
    hypothesis("K lies on the diameter AC, strictly inside", on_line(k, Line.through(a, c))
               and k != a and k != c)
    line(a, c, "the diameter AC")

    across = _parallel_through(k, a, b)
    along = _parallel_through(k, a, d)
    e = posit(meet_one(along, Line.through(a, b)), "E")
    f = posit(meet_one(along, Line.through(d, c)), "F")
    g = posit(meet_one(across, Line.through(b, c)), "G")
    h = posit(meet_one(across, Line.through(a, d)), "H")

    # The whole and each parallelogram about the diameter answer to I.34.
    because(prop_I_34, a, b, c, d)
    because(prop_I_34, a, e, k, h)
    because(prop_I_34, k, g, c, f)

    claim("the diameter bisects the whole parallelogram", "I.34",
          _area(a, b, c) == _area(a, c, d))
    claim("and it bisects each of the parallelograms about it", "I.34",
          _area(a, e, k, h) == 2 * _area(a, e, k) and _area(k, g, c, f) == 2 * _area(k, g, c))
    claim("so the complements EBGK and HKFD are equal", "C.N.3",
          _area(e, b, g, k) == _area(h, k, f, d))
    return Out(complements=((e, b, g, k), (h, k, f, d)))