Coordinates of commom point of lines determined by four points with determinants

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Given x,y coordinates of four points, say A, B, C and D, is there a form with determinants for the coordinates of commom point of lines AB and CD?

Each line equation is already given by a determinant, and the solution of the linear system is also given by a determinant, with Cramer's Rule, so I wonder if there is a nice form for x,y of the commom point, as my final formula is giant.