Small triangles within a large triangle

Geometry Level 4

Find the minimum value of $S$ such that the following statement is true:

Given any configuration of 100 points which satisfies the condition that any 3 points determine a triangle of area $\leq 1$ , there exists a triangle of area $S$ which encloses all these points.

The answer is 4.00.

1 solution

Ossama Ismail
Feb 6, 2017

Choose any 3 points $XYZ$ so that the area of $\triangle X Y Z <= 1$ .
Draw parallels $AB \parallel XZ, BC \parallel YX, AC \parallel YZ$ and construct $\triangle ABC$ . The area of $\triangle ABC$ will be $\leq 4$ .
Now we need to show that $\triangle ABC$ contains the 100 points which satisfy that any triplets have an area $\leq 1$ .
Assume that a point $D$ is outside the $\triangle X Y Z$ , see the above figure. The triangle area of $\triangle DYZ \ \$ is $\ > \$ area of $\triangle XYZ$ . This contradicts the assumption. Then the maximum area of $\triangle A B C$ doesn't exceed $4$ annd all other points are inside or at the borders of $\triangle ABC$ .

The number of points in this problem has no meaning and this solution is valid for any number of points satisfying the given condition.

Moderator note:

This solution shows that there exists a triangle of area 4 that encloses all these points.

However, it has not shown that 4 is the minimum. What we likely have to do, is show that for every $B < 4$ , there exists a configuration of 100 points which cannot be enclosed by a triangle of area B.

First, in order to prove that something is a maximum/minimum, we have to show that
1. It is an upper bound.
2. It can be achieved.

In your solution, you have shown that 4 is an upper bound. However, I do not believe that it can be achieved.

In addition, a sentence of "The minimum area of $ABC$ is $\leq 4$ " while true, doesn't exclude possibilities like "The minimum area of $ABC$ is $\leq 5$ ".

Calvin Lin Staff - 4 years, 4 months ago

Can you disprove it?

The points A, B, and C are selected to satisfy the condition that area of any new triangle's area will not exceed 1. Any other choices will produce a contradiction.

Ossama Ismail - 4 years, 4 months ago

I agree that you have proven 4 is an upper bound. However, 5 is also upper bound, so why isn't 5 a valid answer to this problem?

The issue is that you want the minimum area of ABC, as opposed to "any upper bound on the area of ABC".

Conversely, can you show that "the minimum area of ABC is strictly greater than 3? 3.9? 3.99?"

Calvin Lin Staff - 4 years, 4 months ago

Note that it is not true "Given any triangle of area 4, and any 100 points in the triangle, any set of 3 points would have an area of at most 1."

Also, note that "minimum area of ABC" could be 0. IE The conditions in your question are satisfies by the degenerate triangle.

Hence, a lot of the phrasing needs to be cleaned up.

Calvin Lin Staff - 4 years, 4 months ago

Small triangles within a large triangle

The answer is 4.00.

1 solution

Moderator note:

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