Problem 6

In the plane, let a set of

n

points,

n\\ge3

, be given, and join every pair by a segment. Let

d

be the length of the longest segment. A diameter is any joining segment of length

d

. Prove that the number of diameters is at most

n

Step 1 of 4: Prove the crossing lemma

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In plain words

Maximum-length chords cannot sit apart: one cross-connection would become even longer.

PQ=RS=d,\\quad PQ\\cap RS=\\varnothing\\Longrightarrow\text{some cross-distance is }>d.

Detailed analysis

Suppose diameter segments $PQ$ and $RS$ are disjoint. At the endpoint $Q$ , the angle between $QP$ and $QR$ is less than $90^\circ$ ; otherwise $PR>d$ by the cosine law. The same argument at each endpoint makes all four angles of the quadrilateral acute, which is impossible. Hence any two diameter segments intersect or share an endpoint.