It is known that the theorems of Desargues and Steiner are the main theorems of projective geometry. In all sources, the proofs of these theorems are given by axiomatic methods. Since the axiomatic method consists of theory, it will not be easy for everyone to understand it. The article presents analytical proofs of the theorems of Desargues and Steiner in projective coordinates. When introducing the projective coordinate system, a set of three vertices was used. A single point is taken as the center of the projection. When proving Desargues' theorem, he effectively used the equation of a projective line on a projective plane, as well as homogeneous coordinates. The equations of second-order curves on the projective plane are derived as the common points of two central families of straight lines.