Nuprl Lemma : Steiner-LehmusTheorem
∀e:EuclideanPlane. ∀a,b,c,x,y:Point. (a # bc ⇒ a-x-b ⇒ c-y-b ⇒ ay ≅ cx ⇒ cay ≅a bay ⇒ acx ≅a bcx ⇒ ab ≅ cb)
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
geo-tri: Triangle(a;b;c),
top: Top,
geo-cong-angle: abc ≅a xyz,
geo-strict-between: a-b-c,
geo-lsep: a # bc,
oriented-plane: OrientedPlane,
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
hp-angle-sum: abc + xyz ≅ def,
geo-out: out(p ab),
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
true: True,
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
rev_implies: P ⇐ Q,
geo-parallel-points: geo-parallel-points(e;a;b;c;d)
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y:Point.
(a \# bc {}\mRightarrow{} a-x-b {}\mRightarrow{} c-y-b {}\mRightarrow{} ay \mcong{} cx {}\mRightarrow{} cay \mcong{}\msuba{} bay {}\mRightarrow{} acx \mcong{}\msuba{} bcx {}\mRightarrow{} ab \mcong{} cb)
Date html generated:
2020_05_20-AM-10_45_19
Last ObjectModification:
2020_01_27-PM-10_05_51
Theory : euclidean!plane!geometry
Home
Index