By Ramis E., Deschamps C., Odoux J.

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Ii) Any three distinct points, no two of which are incident with a common line, are incident with exactly one common circle. (iii) If x and y are distinct noncollinear points of £ and if C is a circle incident with x but not with y, then there is just one circle C incident with x, y and having only x in common with C (two circles having exactly one point x in common will be called tangent circles at x). (iv) There exist a point x and a circle C with x IC; each circle is incident with at least three points.

Hence the point x is antiregular. Conversely, assume that x is antiregular. 1, s is odd. 3 The point (oo) of any GQ T 2 (C) of order q, with q odd, is antiregular. Proof. 1 the point (oo) of T 2 (C) is coregular, hence by the previous theorem it is antiregular. 4 Antiregularity and Laguerre Planes This section will be about the connection between antiregular points, affine planes and Laguerre planes. 1 Let x be an antiregular point of the GQ ~~
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~~Generalized quadrangles were introduced by Tits [217] in the appendix of his celebrated work on triality. If x and L are as in (iii), then we will denote the unique point y on L collinear with x also by projLa;, and call it the projection ofx onto L. Dually we define proj x L •= M. We generalize this to points z on L by putting z = proj L z, if zIL, and to lines M concurrent with L by denoting the intersection point proj L M. The dual notation is also used. l 2 Chapter 1. Generalized Quadrangles The integers s and t are the parameters of the GQ and S is said to have order (s, t); if s = t, S is said to have order s. ~~