Suppose ten points are placed at random inside a triangle whose sides are all of length three. Show that there must be at least two points no more than a distance one apart. Be sure to consider all possible cases.

So what i did was...

dividing the triangle into 4 smaller equilateral triangles like this:

A

AVA

(hope you understand what i was trying to draw out) and each length has length 1.

And I used the pigeohole priciplae. which says if there are n+1 pigeons in n pigeon holes, at least one hall must contain 2 or more pigeon.

so isn't it obvious that if there are 10 points placed in the triangle, obviously there will be at least 2 points which has a length less than one??

Am I being tricked here, and what does it mean by "be consider to show all cases"???