Imagine a multiplication table (like the one below, except it continues on forever):

1 2 3 4
+---+---+---+---+---
1| 1 | 2 | 3 | 4 |...
+---+---+---+---+---
2| 2 | 4 | 6 | 8 |...
+---+---+---+---+---
3| 3 | 6 | 9 | 12|...
+---+---+---+---+---
4| 4 | 8 | 12| 16|...
+---+---+---+---+---
|...|...|...|...|...

Find three of the same number in a straight line somewhere within the table. If this is not possible, show why not.

It is not possible if we are talking about the center of the three squares being in exactly a straight line.

Consider the graph in an x-y coordinate system of xy = k.

For a given value of k > 0, this is a hyperbola, and it clearly does
not have three colinear points with real values. In particular,
it does not have three colinear points, all of whose (x,y) values are
integers. Therefore, the multiplication table does not have three
of the same number in a straight line.

However, if we are less rigorous, then you can draw a straight line
which passes through all three squares marked with a "4" in tomarken's
picture.