Book II · Proposition 5

II.5

If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.Heath, 1908

Euclid's tool for solving quadratics: ab + ((a-b)/2)^2 = ((a+b)/2)^2.

Every step, checked

• B cuts AC unequallygiven
• the two sections are distinctgiven
• D bisects ACby I.10
• the drawn rectangle is contained by the unequal segmentsby Def.22
• and the drawn square is the square on the halfby I.46
• the rectangle on the unequal segments, with the square on the piece between the sections, equals the square on the halfby I.43, I.46

What it needs, and what needs it

Needs: I.10 I.43 I.46

Used by: II.6 II.14 III.35 VI.27 VI.28 X.17

Rests on: C.N.1, C.N.2, C.N.3, C.N.4, C.N.5, Def.10, Def.15, Def.22, Def.4, Post.5

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 (×9)continuity
• a line-circle intersection is used, though no postulate asserts that a line meets a circle (×14)continuity
• the argument reads a 'between' fact off the figureorder

The proposition as code

@proposition(
    "II.5",
    THEOREM,
    sample=_adjacent_segments,
    note="Euclid's tool for solving quadratics: ab + ((a-b)/2)^2 = ((a+b)/2)^2.",
)
def prop_II_5(a: Point, b: Point, c: Point) -> Out:
    """AC is bisected at D and cut unequally at B."""
    hypothesis("B cuts AC unequally", between(a, b, c))
    middle = posit(prop_I_10(a, c).midpoint, "D")
    hypothesis("the two sections are distinct", middle != b)
    outline(a, c, close=False)

    # The rectangle on the unequal segments stands above the line, the square on
    # the half below it, so the two areas being compared can be seen at once.
    rectangle = _rectangle(a, b, _across(b, c), "E", "F")
    square = _rectangle(middle, c, _across(c, middle), "H", "G")

    first, second = length(a, b), length(b, c)
    half, offset = length(a, middle), length(middle, b)
    because(prop_I_46, middle, c)
    # Euclid's gnomon: the square on the half, cut by the diameter, has the
    # piece between the sections standing at the corner, and I.43 makes the two
    # complements equal. K is that corner, taken along the diameter.
    _fraction = offset / half
    _corner = posit(Point(square[0].x + _fraction * (square[2].x - square[0].x),
                          square[0].y + _fraction * (square[2].y - square[0].y)), "K")
    because(prop_I_43, square[0], square[1], square[2], square[3], _corner)

    claim("D bisects AC", "I.10", eq_len(a, middle, middle, c))
    claim("the drawn rectangle is contained by the unequal segments", "Def.22",
          _area(*rectangle) == first * second)
    claim("and the drawn square is the square on the half", "I.46",
          _area(*square) == half * half)
    claim("the rectangle on the unequal segments, with the square on the piece between "
          "the sections, equals the square on the half", ["I.43", "I.46"],
          _area(*rectangle) + offset * offset == _area(*square))
    return Out(midpoint=middle, rectangle=rectangle, square=square)