 We can apply the pigeonhole principle to geometric problems. It's useful to think of the pigeonholes as sorting bins. For example, what's the minimum number of points you have to place in a 3-inch square to guarantee at least two points are within square root 2 inches of each other? It will introduce a useful strategy in geometry, tessellate. In other words, can you divide the region into similar figures? Suppose we take our 3-inch square and divide it into 9 1-inch squares. Two points in any square must be within square root 2 inches of each other, so if the squares are our bins, then we need at least 10 points. A couple more useful strategies emerge from the following of any four distinct lines in the plane, either to our parallel or all four intersects pairwise. So suppose we start with one line, L1. If we wanted to sort the lines, we could sort them into parallel or intersecting. Since there are three lines remaining but only two possibilities, then one of the bins has to have at least two of the lines. Now if the parallel bin has one line, then there are two parallel lines and we're done. But remember to always ask, what if it's not? And in this case, if it's not, then the remaining three lines have to intersect the given line. And so all four lines intersect pairwise. Or do they? Remember, if you don't find the flaws in your reasoning, someone else will. And in this case we know that the three lines intersect the given line, but we don't know if they intersect each other. So we'll introduce another strategy, Lather Rinse Repeat. So let's pick any line L2 by assumption L2 intersects L1 and the remaining lines are either parallel or the intersect L2. If one of them is parallel, we're done. Because now we have two parallel lines. If not, consider L3. It intersects L2. It also intersects L1 because it wasn't parallel. So L4 is either parallel to L3 and we're done. If L4 isn't parallel, it also intersects L2 and L1. So all four lines intersect pairwise. Lather Rinse Repeat strategy works on larger cases. So for example, if we have seven lines in the plane, we know that either three are parallel or four intersect pairwise. So again, let's consider any one of them. The remaining lines are either parallel or they intersect L1. Since there are two bins and six remaining lines, at least one bin must have two of the lines. If the parallel bin has two or more lines, we're done because that will give us three parallel lines. So there are at least five lines that intersect L1. So there might be one parallel line. We'll ignore it. Essentially, we're only going to rely on the lines whose properties we can guarantee. Now consider one of the remaining intersecting lines, L2. Either the remaining four lines are parallel or they intersect L2. And remember at this point all of the lines were considering intersect L1. So again, if two of the remaining lines are parallel to L2, we're done, so it means that at least three intersect L2. And again, the line that might be parallel will ignore. So now consider one of those lines, L3. Again, we're ignoring all lines parallel to L1 and L2, so L3 has to intersect both L1 and L2. Again, the remaining two lines are either parallel or they intersect L3. If they're both parallel, we're done. So at least one line intersects L3. And again, it can't be a line parallel to L1 or L2. So this line must also intersect L2 and L1. So there are four lines that intersect pairwise.