By H.S.M. Coxeter

In Euclidean geometry, buildings are made with ruler and compass. Projective geometry is easier: its buildings require just a ruler. In projective geometry one by no means measures whatever, as a substitute, one relates one set of issues to a different through a projectivity. the 1st chapters of this booklet introduce the real strategies of the topic and supply the logical foundations. The 3rd and fourth chapters introduce the recognized theorems of Desargues and Pappus. Chapters five and six utilize projectivities on a line and aircraft, repectively. the following 3 chapters boost a self-contained account of von Staudt's method of the idea of conics. the fashionable technique utilized in that improvement is exploited in bankruptcy 10, which offers with the easiest finite geometry that's wealthy sufficient to demonstrate the entire theorems nontrivially. The concluding chapters exhibit the connections between projective, Euclidean, and analytic geometry.

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**Example text**

1. Regarding ABPCSA1 as a hexagon whose six vertices lie alternately on two lines, what can be said about the intersections of pairs of "opposite" sides of this hexagon? 3. 2A. 4. If H(AB, CD) then ABCD X BACD. 12 tells us that there is only one projectivity ABC n A'B'C' relating three distinct points on one line to three distinct points on the same or any other line. 1A) shows how, when the lines AB and A'B' are distinct, this unique projectivity can be expressed as the product of two perspectivities whose centers may be any pair of corresponding points (in reversed order) of the two related ranges.

This is naturally denoted by (AA)(BB). The harmonic conjugate of C with respect to any two distinct points A and B may now be redefined as the mate of C in the involution (AA)(BB). 42 Any point is its own harmonic conjugate with respect to itself and any other point. EXERCISES 1. If ABCD A BACD then H(AB, CD). ) 2. If a hyperbolic projectivity relates two points that are harmonic conjugates with respect to the invariant points, it must be an involution. 3. If H(AB, MN) and H(A'B', MN), then MN is a pair of the involution (AA')(BB').

Conversely, if we are given the triangle PQR and any line s, not through a vertex, we can define A, B, C to be the harmonic conjugates of the points D, E, Fin which s meets the sides QR, RP, PQ. 31, the three lines PA, QB, RC all pass through a point S, which is the trilinear pole of s. 30 THE PRINCIPLE OF DUALITY EXERCISES 1. What happens if we stretch the definitions of trilinear pole and trilinear polar to the forbidden positions of S and s (namely S on a side, or s through a vertex)? 2. 4A) the trilinear pole of s with respect to the triangle ABC.