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I am working on this problem for some time, and I am not able to finish the argument:

There is a finite number of points in the plane, such that every triangle has area at most 1. Prove that the points can be covered by a rectangle of area 2.

My progress so far:

Consider the two points furthest apart, A and B. Draw lines through A and B, perpendicular to AB. Then every point has to be between the two lines, since AB has maximal length among our pair of points. Now take the points C and D, furthest apart from the line AB, on each of the two halfplanes determined by AB. Every other point in the plane has to be between the two lies through C and D, parallel to AB .

The rectangle determined by the four lines gives us an area of at most 4, but I feel like this argument can be improved somehow to give us the required area of 2.

Thank you! (Edit: thank you for the advice on posting problems here )

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Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many would consider your post rude because it is a command ("Prove..."), not a request for help, so please consider rewriting it. –  Zev Chonoles Jul 4 '12 at 19:13

4 Answers 4

If the theorem is true, a circle that is approximated by a polygon of sufficiently enough vertices verifies the theorem too.

The radius of the circle is $r$. The rectangle is a square and has area $4 r^2$.

The triangle of greatest area in a circle is equilateral, so has area $3/2 \times r \times \sqrt{3}/2 \times r$.

If the theorem is true, $4r^2 \leq 2 \times (3/2 \times r \times \sqrt{3}/2 \times r)$.

So $16 \leq 27 /2$. Contradiction.

The theorem seems to be false.

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An even simpler counterexample would be the corner points of any non-rectangular parallelogram of area 2, e.g. $(0,0)$, $(0,1)$, $(2,2)$ and $(2,1)$. Any triangles formed from those points must have half the area of the parallelogram, i.e. 1, but as the parallelogram is not a rectangle, any rectangle that covers it must have an area greater than 2.

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In the arrangement of four points below, each of the $4$ triangles has area $1$.

$\hspace{7mm}$enter image description here

As $L\to\infty$, it seems that the smallest rectangle enclosing the $4$ points would have to be about $L\times4/L$, which would give it an area of $4$, not $2$.

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+1 for the picture. –  Ilmari Karonen Jul 4 '12 at 21:38

I believe that the theorem you stated is false, but there's a simple variation of your statement that you can actually prove!

revised theorem: Given a finite number, n, of points in a plane such that any three points form a triangle of area 1 or less, then there is a triangle of area 4 in the plane that contains all $n$ given points.

A quick outline of why this is true:

-We have finitely many points in the plane, and hence finitely many ways to form a triangle. So there is a least upper bound $\alpha$ on the areas of the triangles, and at least one triangle has area $\alpha$. Start by drawing this triangle, which vertices we will call A,B and C, and which side we will call (AB), (BC), and (CA).

-From triangle ABC, we can draw a bigger triangle and in so doing complete the proof. Draw the line (AB)' which is parallel to (AB) and passes through vertex C. Similarly draw (BC)' which is parallel to (BC) and passes through vertex A, and (CA)' which is parallel to (CA) and passes through vertex B.
The intersection of (AB)',(BC)', and (CA)', is a triangle of area $4*\alpha \leq 1 $, which we'll call ABC'.

-The important point is that, by construction, ABC' must contain all n of our original points. Indeed, pick a point D distinct from A, B, C. We could draw the triangles ABD,ACD, or BCD, each of which has area less than or equal to $\alpha$. Keeping in mind that $Area=\frac{bh}{2}$, we see that area(ABD)$\leq$area(ABC) implies that the shortest distance between D and (AB) must be less than or equal to the shortest distance between C and (AB). In other words, D must lie between (AB) and (AB)' or between (AB) and the reflection of (AB)' over (AB). Likewise, the shortest distances between D and (CA) and between D and (BC) must be respectively less than or equal to the shortest distances between B and (CA) and A and (BC).
So D must lie also between (CA) and (CA)' or (CA) and the reflection of (CA)' over (CA), and D must lie between (BC) and (BC)' or (BC) the reflection of (BC)' over (BC). This restriction on where the point D can lie can be shown quite easily to be equivalent to the restriction that D must lie in the triangle ABC'. So there is a triangle of area $4*\alpha$, which is less than or equal to 4 by assumption, which contains every point of our plane.

-Informally, this proves the revised version of the theorem you gave. Sorry it's a couple years late, hope it's still interesting or helpful for somebody.

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