By Derrick Norman Lehmer

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Also, from the similar triangles DSC and BCC', we have CD : CB = SD : B'C. From these two proportions we have, remembering that BA' = BC', AB · CD = −1, AD · CB the minus sign being given to the ratio on account of the fact [26] 28 An Elementary Course in Synthetic Projective Geometry that A and C are always separated from B and D, so that one or three of the segments AB, CD, AD, CB must be negative. 45. Writing the last equation in the form CB : AB = -CD : AD, and using the fundamental relation connecting three points on a line, PR + RQ = PQ, which holds for all positions of the three points if account be taken of the sign of the segments, the last proportion may be written (CB - BA) : AB = -(CA - DA) : AD, or (AB - AC) : AB = (AC - AD) : AD; so that AB, AC, and AD are three quantities in hamonic progression, since the difference between the first and second is to the first as the difference between the second and third is to the third.

24] 26 [25] An Elementary Course in Synthetic Projective Geometry Thus, given that B is the middle point of AC, we may draw any two lines through A, and any line through B cutting them in points N and L. Join N and L to C and get the points K and M on the two lines through A. Then KM is parallel to AC. The bisection of a given segment and the drawing of a line parallel to the segment are equivalent data when linear construction is used. 42. It is not difficult to give a linear construction for the problem to divide a given segment into n equal parts, given only a parallel to the segment.

Trace the lines of the string when the two boys move at the same rate of speed in the two paths but do not start at the same time from the point where the two paths intersect. 6. A ship is sailing on a straight course and keeps a gun trained on a point on the shore. Show that a line at right angles to the direction of the gun at its muzzle will pass through any point in the plane twice or not at all. ) 7. Two lines u and u' revolve about two points U and U' respectively in the same plane. They go in the same direction and at the same rate of speed, but one has an angle a the start of the other.

### An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer

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