By Derrick Norman Lehmer

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**Sample text**

Given the middle point of a line segment, to draw a line parallel to the segment and passing through a given point. 6. A line is drawn cutting the sides of a triangle ABC in the points A', B', C' the point A' lying on the side BC, etc. The harmonic conjugate of A' with respect to B and C is then constructed and called A". Similarly, B" and C" are constructed. [28] 30 An Elementary Course in Synthetic Projective Geometry Show that A"B"C" lie on a straight line. Find other sets of three points on a line in the figure.

5). Join O to A and C. The joining lines cut out on the sides of the quadrangle four points, P, Q, R, S. Consider the quadrangle P, K, Q, O. One pair of opposite sides passes through A, one through C, and one remaining side through D; therefore the other remaining side must pass through B. Similarly, RS passes through B and PS and [19] 20 An Elementary Course in Synthetic Projective Geometry QR pass through D. The quadrangle P, Q, R, S therefore has two opposite sides through B, two through D, and the remaining pair through A and C.

It is also possible for two projective pencils to have the same center. This happens, for example, when two projective point-rows are projected to the same point. Similarly, two projective axial pencils may have the same axis. We examine now the possibility of two forms related in this way, having an element or elements that correspond to themselves. We have seen, indeed, that if B and D are harmonic conjugates with respect to A and C, then the point-row described by B is projective to the point-row described by D, and that A and C are self-corresponding points.