Let Θ be the Baby Monster graph which is the graph on the set of {3,4}-transpositions in the Baby Monster group B in which two such transpositions are adjacent if their product is a central involution in B. Then Θ is locally the commuting graph of central (root) involutions in ~2E_6(2). The graph Θ contains a family of cliques of size 120. With respect to the incidence relation defined via inclusion these cliques and the nonempty intersections of two or more of them form a geometry ε(B) with diagram c.F_4(t): for t = 4 and the action of B on ε(B) is flag-transitive. We show that ε(B) contains subgeometries ε(~2E_6(2)) and ε(Fi_(22)) with diagrams c.F_4(2) and c.F_4(1). The stabilizers in B of these subgeometries induce on them flag-transitive actions of ~2E_6(2) : 2 and Fi_(22) : 2, respectively. The geometries ε(B), ε(~2E_6(2)) and ε_(Fi_(22)) possess the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2 and (b) three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The paper addresses the classification problem of c.F_4(t)-geometries satisfying (a) and (b). We construct three further examples for t = 2 with flag-transitive automorphism groups isomorphic to 3·~2E_2(2) : 2, E_6(2) : 2 and 2~(26).F_4(2) and one for t = 1 with flag-transitive automorphism group 3·Fi_(22) ; 2. We also study graph of an arbitrary (non-necessary flag-transitive) c.F_4(t)-geometry satisfying (a) and (b) and obtain a complete list of possibilities for the isomorphism type of subgraph induced by the common neighbours of a pair of vertices at distance 2. Finally, we prove that ε(B) is the only c.F_4(4)-geometry, satisfying (a) and (b).
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