Question

3). 12. Recall that the centroid of a triangle with vertices (11; yı), (22:42), (13,) € RP is the point (1 , Given a set of 20 points in RP with integer coordinates, prove that three of them will form a triangle whose centroid has integer coordinates. Hint: work mod 3 and use the pigeonhole principle. Distribute the points ( yi) into 9 boxes depending on r, mod 3 and yi mod 3.

Answer 1

5 There are three possibilities for any integer when divided by 3.. i.e. n o (mod 3) nel (mod 3) n = 2 (mod 3) Thus, both ; and y. have 3 options each. lit o orł ost Я; е б олт ол 2 We define the pigeonwholes by remainder of the numbers when divided by 3. , 8 | 1 2 115,0) 10,7) / 10,2) T (1,0) |(1,7)|(1,7) ī (2,) (3,7)|(2,7) since, we are picking 20 points we will have | 207 -3 points in the same pigeon whole. .:. There will be three points (x;"y:) , Vi=1,2,3 such that, (Oxi, y:) elo, 0) lie. 2,6 0,72 , aztő le. 21; E O (mod 3) Hi=1,2,3 => 21+ H2 + 32 = 0.1 mod 3) i.e. Nitka +23 is an integer. Similarly, y;€ Ō, il. Y; = 0 (mod 3), Vi= 1,2,3 => Yot Y2+ Y3 = 0[mod 3) i.e. Yrit Pzt Yz is an integer 30. the centeroid of the turiangle formed by Ex-yo), pse 2:42):183143) will have integer co-ordinate (proved]

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