A given trapezoid A1B1B2A2 with bases (parallel sides) A1B1, A2B2 specifies a central dilation or homothecy of an affine space (mapping A1 on A2, B1 on B2), iff the other two sides A1A2, B1B2 are not parallel. In this case, the latter sides meet in a point Z, which is the center of the dilation. Given this center and a pair (B1, B2) of a point and its image, the image of an arbitrary point C1 not on the line B1B2 can be constructed as shown in the picture.
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KlioKlein