Close to Triangular

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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 triangles made.

The 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 triangle.
four triangles through 3 points

Preveina went on to show some examples of configurations of four and five points where a triangle could be drawn.

To consider whether all configurations are possible, consider the set of points below:

four points
Can you find a way to draw a triangle passing through all four points? Can you convince yourself it is impossible?