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Invertibility of Toeplitz operators on the Bergman space and the related Douglas problem are long standing open problems. In this paper we study the invertibility problem under the novel geometric condition on the image of the symbols, which relaxes the standard positivity condition. We show that under our geometric assumption, the Toeplitz operator $T_\varphi$ is invertible if and only if the Berezin transform of $|\varphi|$ is invertible in $L^{\infty}$. It is well known that the Douglas problem is still open for harmonic functions. We study a class of rather general harmonic polynomials and characterize the invertibility of the corresponding Toeplitz operators. We also give a number of related results and examples., Comment: The authors are in the possession of pdf-slides of a talk in a workshop in Wuhan University on Dec. 1st, 2024, where Mingjin Li and J.Long of Guizhou Normal University, China, are claiming the ownership of the main results of our paper with the same notation and the same results. Such claims are utterly false and should be given no attention