Definition contraction a b k := k *: (b - a) + a.

Lemma ratio_bet a b c k1 k2 k3 :
0 < k1 -> 0 < k2 -> k1 < 1 -> 0 < k3 -> k3 < k1+k2-k1*k2 ->
b - a == ((k1+k2-k1*k2)/k3)^-1 *: (c - a) -> bet a b c.