Distance between points in a triangle

[Yankel]

Hello all,

Below there is a problem:

There are five points inside an equilateral triangle of side length 2. Show that at least two of the points are within 1 unit distance from each other.

I have plotted such a triangle using "Geogebra", and attaching the picture.

I know that if I create another equilateral triangle within the original one, I get four triangles. Then, according to the pigeonhole principle, with 5 points (pigeons) and 4 triangle (holes), at least two points will be in the same triangle.

My questions are:

1) What is the geometrical reasoning for claiming that two points within a triangle will have a distance of 1 units max ? I couldn't prove it.
2) What happens if a point in the bigger triangle happens to be on the edge of the inner black triangle? Doesn't it disproof the theory ?

Thank you in advance !

View attachment 7554

 

[kaliprasad]

Hello all,

Below there is a problem:

There are five points inside an equilateral triangle of side length 2. Show that at least two of the points are within 1 unit distance from each other.

I have plotted such a triangle using "Geogebra", and attaching the picture.

I know that if I create another equilateral triangle within the original one, I get four triangles. Then, according to the pigeonhole principle, with 5 points (pigeons) and 4 triangle (holes), at least two points will be in the same triangle.

My questions are:

1) What is the geometrical reasoning for claiming that two points within a triangle will have a distance of 1 units max ? I couldn't prove it.
2) What happens if a point in the bigger triangle happens to be on the edge of the inner black triangle? Doesn't it disproof the theory ?

Thank you in advance !

Suppose there are 2 points in the triange CDE.

the largest distance between any 2 points in CDE (The points in the edges included) is 1.

to prove the same draw an arc with centre C and distance 1
DE line segment shall be in the arc and any poins in DE shall be < 1 (unless end point D or E for which distance = 1)
so maximum distance is 1