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In the study of classical obstacle problems, it is well known that in manyconfigurations the free boundary intersects the fixed boundary tangentially.The arguments involved in producing results of this type rely on the linearstructure of the operator. In this paper we employ a different approach andprove tangential touch of free and fixed boundary in two dimensions for fullynonlinear elliptic operators. Along the way, several $n$-dimensional results ofindependent interest are obtained such as BMO-estimates, $C^{1,1}$ regularityup to the fixed boundary, and a description of the behavior of blow-upsolutions.

Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions