1. Let P = {Pl,P2,... ,Pn} be a set of n points in the Euclidean plane and S be the set of connecting lines, which join these points. The straight line s E S is called straight line of order i when Is 5 P[ = i. For a point-set P we denote Si(P) the set of straight lines of order i and ti = [ Si(P)[ 9 Then n (1.1) t = E ti i=2 where t = [S[, is true. The connecting line s 6 S is called ordinary when IsnPI =2. For a given pointset P let C be a set of circles determined by points of P. The circle c E C is called circle of order i when [e n PI = i. The set of circles of order/is denoted by C~(P) and ki = ]Ci(P)I. Then Tt (1.2) k : i=3 where k = IC[, is true. A circle c E C ordinary when [c n PI = 3.2. Now we formulate our main result in following two theorems. TI~EOREM 2.1. Let P be a set of n >= 4 points in the Euclidean plane, not all on a circle or a straight line. Let pj be an arbitrary point of a set P.Then P determines at least 15(n-1)/133 circles containing exactly three points of P, one of which is pj. Elliott [6] has proved IC3(P, pj)l >= 2(n-1)/21, where C{(P, pj)is the set of the circles of order { "" determined by P and passing through pj. ~TIIEOItEM 2.2. Let P satisfy the hypotheses of Theorem 2.1. Then k3 >= > ~n(n-1)/133. From the above mentioned theorems further results follow. For the upper bound of the number of circles of order m Jucovi~ [10] has proved the estimate (2.1) < 1
ABSTRACT. We show that for any convex body K ~ E 2 there exists a triangle T such that T c K ~ (3(x/i7-1)/4)T, where 2T is a suitable homothetic copy of T with ratio 2. As a corollary we show that if (Ki) are homothetic copies of a given convex body K c E 2 with area V(K) = 1, then the condition E V(K~) >~ 9(9-x/~)/4 is sufficient for the existence of a translative covering of K by (Ki).
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