Book I · Proposition 39

I.39

Equal triangles which are on the same base and on the same side are also in the same parallels.Heath, 1908

Every step, checked

• C and D are on the same side of ABgiven
• the triangles ABC and ABD are equalgiven
• were CD not parallel to AB, a parallel through C would cut BD and I.37 would make a part equal the wholeby I.37

What it needs, and what needs it

Needs: I.31 I.37

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: 13 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 meetcontinuity
• a line-circle intersection is used, though no postulate asserts that a line meets a circlecontinuity
• the argument reads a 'same_side' fact off the figureorder

The proposition as code

@proposition(
    "I.39",
    THEOREM,
    sample=_equal_triangles_same_base,
)
def prop_I_39(a: Point, b: Point, c: Point, d: Point) -> Out:
    base = Line.through(a, b)
    hypothesis("C and D are on the same side of AB", same_side(c, d, base))
    hypothesis("the triangles ABC and ABD are equal", eq_area((a, b, c), (a, b, d)))
    outline(a, b, c)
    outline(a, b, d)
    line(c, d)

    # The parallel Euclid supposes drawn through C is drawable; where it cuts
    # BD is the point his argument compares against, and I.37 speaks of it.
    alongside = prop_I_31(c, a, b).parallel
    cut = meet_one(alongside, Line.through(b, d))
    because(prop_I_37, a, b, c, cut)

    claim("were CD not parallel to AB, a parallel through C would cut BD and I.37 would "
          "make a part equal the whole", "I.37", parallel(Line.through(c, d), base))
    return Out()