LetC={Cα}α∈A∈[1;∞)A,A-index set. A quasi-triangular space(X,PC;A)is a setXwith familyPC;A={pα:X2→[0,∞), α∈A}satisfying∀α∈A ∀u,v,w∈X {pα(u,w)≤Cα[pα(u,v)+pα(v,w)]}. For anyPC;A, a left (right) familyJC;Agenerated byPC;Ais defined to beJC;A={Jα:X2→[0,∞), α∈A}, where∀α∈A ∀u,v,w∈X {Jα(u,w)≤Cα[Jα(u,v)+Jα(v,w)]}and furthermore the property∀α∈A {limm→∞pα(wm,um)=0} (∀α∈A {limm→∞pα(um,wm)=0})holds whenever two sequences(um:m∈N)and(wm:m∈N)inXsatisfy∀α∈A {limm→∞supn>mJα(um,un)=0andlimm→∞Jα(wm,um)=0} (∀α∈A {limm→∞supn>mJα(un,um)=0andlimm→∞Jα(um,wm)=0}). In(X,PC;A), using the left (right) familiesJC;Agenerated byPC;A(PC;Ais a special case ofJC;A), we construct three types of Pompeiu-Hausdorff left (right) quasi-distances on2X; for each type we construct of left (right) set-valued quasi-contractionT:X→2X, and we prove the convergence, existence, and periodic point theorem for such quasi-contractions. We also construct two types of left (right) single-valued quasi-contractionsT:X→Xand we prove the convergence, existence, approximation, uniqueness, periodic point, and fixed point theorem for such quasi-contractions. (X,PC;A) generalize ultra quasi-triangular and partiall quasi-triangular spaces (in particular, generalize metric, ultra metric, quasi-metric, ultra quasi-metric,b-metric, partial metric, partialb-metric, pseudometric, quasi-pseudometric, ultra quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra quasi-gauge, and partial quasi-gauge spaces).