lineList = {{p0, p5, p8, p10}, {p0, p3, p7, p9}, {p0, p2, p4, p6}, {p0, p1}, {p1, p6, p9, p10}, {p1, p4, p7, p8}, {p1, p2, p3, p5}} ;

<< DiscreteMath`Combinatorica`

possibleTri = KSubsets[Union[Flatten[lineList]], 3] ;

test1T[t_] := Length[Intersection[t, #]] & /@ lineList

myTris = Select[possibleTri, Count[test1T[#], 2] == 3 &]

{{p0, p1, p10}, {p0, p1, p2}, {p0, p1, p3}, {p0, p1, p4}, {p0, p1, p5}, {p0, p1, p6}, {p0, p1, p7}, {p0, p1, p8}, {p0, p1, p9}, {p0, p10, p6}, {p0, p10, p9}, {p0, p2, p3}, {p0, p2, p5}, {p0, p3, p5}, {p0, p4, p7}, {p0, p4, p8}, {p0, p6, p9}, {p0, p7, p8}, {p1, p10, p5}, {p1, p10, p8}, {p1, p2, p4}, {p1, p2, p6}, {p1, p3, p7}, {p1, p3, p9}, {p1, p4, p6}, {p1, p5, p8}, {p1, p7, p9}}

Length[myTris]

27

Converted by Mathematica  (December 18, 2004)