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There are four pairs of values modulo two, so if you have five points some two of them must be the same modulo two. Say these points are A and B. Then A + B = (
by nilkn 14y ago
There are four pairs of values modulo two, so if you have five points some two of them must be the same modulo two. Say these points are A and B. Then A + B = (even, even), and so (A + B) / 2 has integer coordinates.
- tantalor 14y agoThat was tough to follow, let me restate, Let the two points be (x1, y1) and (x2, y2) such that x1 % 2 == x2 % 2 and y1 % 2 == y2 % 2, i.e., have the same parity. Then x1 + x2 = 2 * x3 (is even) and y1 + y2 = 2 * y3 (is even), and (x3, y3) is an integer point. Given five points, two such points must exist because of the pigeon hole principle.
- olivier1664 14y agoSo that means that there will be at least on line which the center is an integer point.
- jonsen 14y agoThere are four possible coordinate parities (even, even), (even, odd), (odd, even), (odd, odd) Among five points two must have the same parity. Between two points of same parity the difference has the parity (even, even). As (even, even) is divisible by two there's an integer midpoint.
- nazgulnarsil 14y agotrue, you can always make one of the 4 lines have a slope that is even/even. the point composed of x/2,y/2 will lie on this line.