given five points in the plane with integer coordinates, show the midpoint between at least one pair of the?

Question

points has integer coordinates.

Answer 1

Start with a point. Let's imagine it is on even coordinates (even, even)

In order to place another point, where the midpoint is a fractional amount, it must be an odd difference vertically or horizontally, or both. In other words, you can't have another (even, even) point. You must pick something different, like (even, odd).

After that, you now can't pick (even, even) or (even, odd) so you must pick a different pairing, say (odd, even).

And the fourth point can't repeat any of the prior combinations, so you have to pick (odd, odd).

As you can see there are 4 combinations (odd, odd), (even, even), (odd, even), (even, odd).

The fifth point will have the same pattern of even and odd as one of the prior points. As you remember the difference between two even numbers is an even number. And the difference between two odd numbers is an even number. Half of two even numbers, results in integer coordinates.

So there is no way to pick 5 points that don't have at least one pair with integer coordinates on their midpoint.

Answer 2

This is an application of the "pigeonhole principle".

If you have a point in the plane with integer coordinates, the x-coordinate can be either even or odd, and the y-coordinate can be either even or odd.

(odd, even)
(even, odd)
(even, even)
(odd, odd)

If you add an even to an even, the result must be even, so you can divide it by 2 and get an integer result.
If you add an odd to an odd, the result also must be even, so the same thing happens.

So to guarantee the midpoint has integer coordinates, you have to find two points where both the x-coordinates are odd or even, and both the y-coordinates are odd or even.

If you pick only 4 points, like the list above, you can't guarantee that any two of them will match.

But with the 5th point, you have no choice; you have to pick one that matches one of the list above.

So then when you take the midpoint of that point and the point it matches, the midpoint will have integer coordinates.

Answer 3

So the midpoint of two points (X1,Y1) and (X2,Y2) is (X1+X2 / 2, Y1+Y2 / 2), so X1+X2 and Y1+Y2 must both be even so that when divided by 2, the quotient is still an integer. Two numbers' sum is even if both numbers are even or odd, but not if one is even and one is odd. IE If two numbers have the same parity, their sum is divisible by two.

Let the 5 points given be (a1,b1).. (a5,b5). There are 4 possible combinations of parity in a point: (even, even), (even, odd), (odd, even), (odd, odd). Since there are five points, at least one of the 4 combinations must be repeated in two ordered pairs. Then these two ordered pairs have both X and Y of the same parity, so their midpoint will have integer coordinates.

Answer 4

Midpoint formula.

((x sub2-x sub1)/2, y sub2-y sub 1)/2)