Nuprl Lemma : opposite-side-congruent-diagonals-midpoint
∀e:BasicGeometry. ∀A,B,C,D,P:Point.
((¬Colinear(A;B;C)) ⇒ B # D ⇒ AB ≅ CD ⇒ BC ≅ DA ⇒ Colinear(A;P;C) ⇒ Colinear(B;P;D) ⇒ {A=P=C ∧ B=P=D})
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-midpoint: a=m=b,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-sep: a # b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
not: ¬A,
false: False,
stable: Stable{P},
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-cong-tri: Cong3(abc,a'b'c'),
geo-five-seg-compressed: FSC(a;b;c;d a';b';c';d'),
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m,
cand: A c∧ B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
rev_implies: P ⇐ Q
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,C,D,P:Point.
((\mneg{}Colinear(A;B;C))
{}\mRightarrow{} B \# D
{}\mRightarrow{} AB \mcong{} CD
{}\mRightarrow{} BC \mcong{} DA
{}\mRightarrow{} Colinear(A;P;C)
{}\mRightarrow{} Colinear(B;P;D)
{}\mRightarrow{} \{A=P=C \mwedge{} B=P=D\})
Date html generated:
2020_05_20-AM-09_57_41
Last ObjectModification:
2020_01_27-PM-10_00_52
Theory : euclidean!plane!geometry
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