Steiner's theorem on a complete quadrilateral consists of 10 interrelated statements, and it was published in the form of a problem or a challenge, that is, without proof, in the 1827./1828. volume of the journal "Annales de Mathematiques Pures et Applique'es", also known as "Journal de Gergonne". The main goal of this paper is to present proofs of all 10 statements. To that purpose, the introductory chapter is a brief recapitulation of numerous notions and facts from Euclidean geometry that will be used in the proofs. Some of those concepts are the Simson-Wallace line, the nine-point circle, the polar circle of a triangle, orthogonal pencils of circles, as well as several topics more related to projective geometry, such as the Desargues' theorem and harmonic quadruple of points. In the second chapter, all 10 statements are proven. Some proofs are grouped according to the contents of the statement or the method used in the demonstration. Steiner starts by observing the four triangles associated to the complete quadrilateral, together with their circumcircles and orthocenters. By noticing some specific sets of collinear and concyclic points, he points out further specific points, lines and circles with particular properties. The last three statements are based on the analysis of a complex structure generated by incenters and centers of excircles of the initial triangles. An examination of the obtained configurations leads to the final statement, describing from a completely different perspective the point which occurred in the first proposition as a common point of all four circumcircles of associated triangles.