It is now well known that in the category of finite polyhedra the fixed point property is not preserved by the operations of suspension, Cartesian product, adjunction along a segment, and join. Thus far none of the examples given have involved polyhedra of dimension 2. It is shown in this paper that two fixed points x x and y y of a self-map of a polyhedron K K can be combined in a certain way if a certain criterion is satisfied by the f f -image of a path from x x to y y . Several corollaries follow, one of which is that if K K is a finite simply connected 2 2 -polyhedron with no local separating points, H 2 ( K ) ≠ 0 {H_2}(K) \ne 0 , and K K has a 2 2 -simplex σ \sigma such that π 1 ( K − Int ⁡ σ , z ) {\pi _1}(K - \operatorname {Int} \sigma ,z) is cyclic, then K K fails to have the fixed point property. This eliminates many 2 2 -dimensional polyhedra from consideration as examples.