By Derrick Norman Lehmer

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**Extra info for An Elementary Course in Synthetic Projective Geometry**

**Example text**

They will cut the circle in four points, which will project to any other point of the curve in four harmonic rays; for, by the theorem concerning the angles inscribed in a circle, the angles involved in the second set of four lines are the same as those in the first set. If, moreover, we project the figure to any point in space, we shall get a cone, standing on a circular base, generated by two projective axial pencils which are the projections of the pencils at S and S'. Cut across, now, by any plane, and we get a conic section which is thus exhibited as the locus of intersection of two projective pencils.

An entirely similar discussion shows that If in two projective pencils the line joining their centers is a self-corresponding ray, then the two pencils are perspectively related. [33] 36 An Elementary Course in Synthetic Projective Geometry 54. A similar theorem may be stated for two axial pencils of which the axes intersect. Very frequent use will be made of these fundamental theorems. [34] 55. Point-row of the second order. The question naturally arises, What is the locus of points of intersection of corresponding rays of two projective pencils which are not in perspective position?

This is a theorem of which frequent use will be made. 5. 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.