By G.B. Gurevich; Translators: J.R.M. Radok and A.J.M. Spencer

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38 GEOMETRIC INTRODucrION CHAP. I 3. Show that any two parallelograms are affinely equivalent; explain the conditions under which two trapezoids will be affinely equivalent; answer the same question for the case of any two quadrangles ABCD and A'B'C'D'. 4. Show that the determinant A . ip,,j of a transformation of affine coordinates (cf. 18)) is equal to the ratio of the orientated areas of the new and old coordinate parallelograms: coordinate parallelograms are parallelograms based on the coordinate vectors e, and e2 starting from the origin of coordinates.

Exercise 2). 4. Establish the Euclidean geometric significance of the cross-ratio {xyzr}, if t is an improper point of the straight line. 5. 3) for the sign of the cross-ratio is retained. 6. In order to generalize the concept of harmonic pairs of points to the case when the points in one or both pairs coincide, the condition {xyzr} _ - I must be rewritten in a form which does not involve a denominator. How is the harmonic property defined in this case? Write the condition for pairs of points to be harmonic in the stated form under the assumption that the points z and t are the first and second coordinate points.

The points with the coordinates (1, 0), (0, 1), (1, 1) are called the first, second and unit coordinate points, respectively; thus, a projective coordinate system is determined by the specification of three coordinate points, where one may select for this role any three distinct points of a straight line. 10) where xI are the coordinates of the transformed point x and . ` are the coordinates, in the same system of projective coordinates, of the point which corresponds to it as the result of the transformation.