if you draw 4 points in a square shape and try to join them all with non-crossing path you can do that, but if you draw 5 points in a shape of pentagon and try to join them it does not work. So, the question is how to prove using one of Euler's equation that its not possible to connect all 5 points with noncrossing path. He also told us that there's 5 vertices,9 edges and 6 faces.
To prove that it is not possible to connect all 5 points in a shape of a pentagon with non-crossing paths, we can use Euler's formula:
V - E + F = 2
where V represents the number of vertices, E represents the number of edges, and F represents the number of faces.
In the given problem, we have 5 vertices, 9 edges, and 6 faces.
Now, let's assume that it is possible to connect all 5 points with non-crossing paths. By doing so, we would create a planar graph with a certain number of edges and faces.
In a planar graph, a face is a region bounded by edges, and a face can be either an internal face or an external face (which extends to infinity).
Since we have a pentagon (which has 5 internal faces) and one external face, our number of faces is 6.
Therefore, we have all the values we need to use Euler's formula:
5 - 9 + 6 = 2
This equation simplifies to:
2 = 2
The equation holds true, indicating that the planar graph we created by connecting the 5 points in a pentagon shape without any crossings has the properties of a planar graph.
Based on Euler's formula, this means that it is possible to connect the 5 points in a pentagon shape with non-crossing paths.
Thus, the information provided is contradictory, and it is possible to connect all 5 points in a pentagon shape with non-crossing paths.