Preveina from Crest Girls' Academy made a start
on this problem:
For three points, there are always infinitely many such triangles
because every time you extend the length of the lines in a triangle
you will be making a new point; by doing this you'll be producing
unique triangles every time. This then leads on having infinity
picture below shows a sequence of triangles - the black lines pass
through two of the points, and a variety of lines can pass through
the third point, extending one of the lines in the original
Preveina went on to show some examples of
configurations of four and five points where a triangle could be
consider whether all configurations are possible, consider the set
of points below:
you find a way to draw a triangle passing through all four points?
Can you convince yourself it is impossible?