The Extreme Principle and Contradiction

The extreme principle is a simple yet very effective way of analyzing many problems. It basically says that, if possible, focus on the "largest" and "smallest" elements of the problem. However, sometimes it is not obvious where this simple insight can be applied, and it requires a lot of creativity to see it.

Contradiction is another powerful tool. When we wish to prove something, we instead assume the opposite, and show that this "contrapositive statement" implies something which we show to be impossible. A simple example of this would be, prove there is no largest integer. Suppose such an integer exists. Call it n. Obviously, we can simply add 1 to that integer to obtain an even larger integer, contradicting that we had the largest one.

Let B and W be a finite set of black and white points, respectively, in the plane, with the property that every line segment that joins two points of the same color also passes through a point of the opposite color. Prove that both sets must lie on a single line segment.

Assume they are not all on the same line segment. Therefore, it must be possible to form at least one or more triangles by drawing all possible line segments. Consider the smallest area possible when connecting three of these points, regardless of what set they are from. Clearly, at least two of the points must be the same color, as we have three points and only two colors. Therefore, there must be another point of the opposite color in between them. However, by using this point and two of the others, we can construct a SMALLER triangle, contradicting that there exists a smallest triangle. This means either one of two things: there are no triangles, or triangles which keep getting infinitely small. Since there are a finite number of points, it is obviously the first one, and if it is impossible to draw a triangle from them, they must all be on the same line. Just as planned.


Consider finitely many points in the plane such that, if we choose any three points A,B,C among them, the area of triangle ABC is always less than 1. Show that all of these points lie within the interior or on the boundary of a triangle with area less than 4. (Korea 1995)

Consider the largest of these triangles, and call it ABC. It is given that [ABC] < style="font-weight: bold;">medial triangle of ABC, which means that A, B, C are the midpoints of that triangle. It is given from basic geometry that [LMN] = 4[ABC] < style="font-style: italic;">P is outside of the triangle. Then we can connect it to two of the vertexes of ABC to create a triangle of area greater than [ABC]. However, this contradicts that [ABC] is the largest triangle. Therefore, all of the points lie inside LMN, and [LMN] < 4. Just as planned.