Roberts's Triangle Theorem

Roberts's triangle theorem, a result in discrete geometry, states that every simple arrangement of $n$ lines has at least $n-2$ triangular faces. Thus, three lines form a triangle, four lines form at least two triangles, five lines form at least three triangles, etc. It is named after Samuel Roberts, a British mathematician who published it in 1889.

Statement and example

The theorem states that every simple arrangement of $n$ lines in the Euclidean plane has at least $n-2$ triangular faces. Here, an arrangement is simple when it has no two parallel lines and no three lines through the same point. A face is one of the polygons formed by the arrangement, but not crossed by any of its lines. Faces may be bounded or infinite, but only the bounded faces with exactly three sides count as triangles for the purposes of the theorem.

One way to form an arrangement of $n$ lines with exactly $n-2$ triangular faces is to choose the lines to be tangent to a semicircle. For lines arranged in this way, the only triangles are the ones formed by three lines with consecutive points of tangency. The other faces of this arrangement are either bounded quadrilaterals, or unbounded. As the $n$ lines have $n-2$ consecutive triples, they also have $n-2$ triangles.

Proof

Branko Grünbaum

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