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(*   Some standard geometry definitions                                       *)
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Definition collinear x y z := x ∨ y ∨ z = 0.
Definition line (x y : point) := x ∨ y.
Definition intersection (x y : vect) := x ∧ y.
Definition concur (l1 l2 l3 : vect) := l1 ∧ l2 ∧ l3 = 0.
Definition online (x y z : point) := y ∨ z <> 0 /\ x ∨ y ∨ z = 0.
Definition online1 (x : point) k1 (y : point) k2 (z : point) :=
x = k1 .* y + k2 .* z :> vect.
Definition inter_lines (x a b c d : point) : Prop :=
x = (a ∨ b) ∧ (c ∨ d) :> vect.

Lemma online_def x y z :
online x y z -> exists pk, online1 x (fst pk) y (snd pk) z.