Book VI · Proposition 5

VI.5

If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.Heath, 1908

The converse of VI.4: proportional sides force equal angles. Together they make similarity a single notion rather than two.

Every step, checked

• neither triangle is degenerategiven
• the sides are proportionalgiven
• the triangles are equiangularby I.8
• and the equal angles are those the corresponding sides subtendby I.8

What it needs, and what needs it

Needs: I.8

Used by: VI.7

Rests on: C.N.1, C.N.3, C.N.4, C.N.5, Def.15, Def.4

Depth: 6 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

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

The proposition as code

@proposition(
    "VI.5",
    THEOREM,
    sample=_similar_triangles,
    note="The converse of VI.4: proportional sides force equal angles. Together "
    "they make similarity a single notion rather than two.",
)
def prop_VI_5(a: Point, b: Point, c: Point, d: Point, e: Point, f: Point) -> Out:
    hypothesis("neither triangle is degenerate",
               not collinear(a, b, c) and not collinear(d, e, f))
    hypothesis("the sides are proportional", similar((a, b, c), (d, e, f)))
    outline(a, b, c)
    outline(d, e, f)

    because(prop_I_8, a, b, c, d, e, f) if congruent_sss((a, b, c), (d, e, f)) else None

    claim("the triangles are equiangular", "I.8",
          eq_angle(b, a, c, e, d, f) and eq_angle(a, b, c, d, e, f)
          and eq_angle(a, c, b, d, f, e))
    claim("and the equal angles are those the corresponding sides subtend", "I.8",
          eq_angle(a, c, b, d, f, e))
    return Out()