Pigeonhole Principle Points in a Triangle

Question

Suppose we have an equilateral triangle with side length $1$. In this equilateral triangle, we place $8$ points either on the boundary or inside the triangle itself. Then what is the maximum possible value for the shortest distance between any two of the points?

An interesting conjecture of Paul Erdos and Norman Oler states that, if $n=\frac{k(k+1)}{2}$ is a triangular number, then the optimal number for $n-1$ points is the same as for $n$ points; this sheds some light on that the answers for $n=10$ and for $n=9$ are the same (and how to find them) but this can hardly help you to find the optimum for $n=8$.

Any thoughts/answers?

Answer 1

I think this is answered by the diagram for $n=8$ at this link.

Answer 2

If you accept the link pointed out by Gerry Myerson then you can add some lines to get this diagram

Your equilateral triangle is the red one; in the diagram, you have to subtract $\sqrt{3}$ twice from the edge of the black equilateral triangle, while the centres of touching circles are $2$ apart. So to get the answer to your question for an equilateral triangle with side length 1 you need

$$\dfrac{2}{(2 + 2 \sqrt{3} + 2\sqrt{33}/3)-2 \sqrt{3}} = \dfrac{\sqrt{33}-3}{8} \approx 0.343$$

compared with $\dfrac{1}{3} \approx 0.333$ for nine or ten points.