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 3 of 4: Show the middle endpoint has degree one

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C\text{ cannot be incident with another diameter }CX;\\quad\text{otherwise }CX\text{ would have to meet both }AB\text{ and }AD.

Detailed analysis

By the crossing lemma, a further diameter $CX$ must meet each of $AB$ and $AD$ , because it cannot be disjoint from either. But with $AC$ lying between $AB$ and $AD$ at their common endpoint $A$ , a single straight segment from $C$ cannot cross both outer segments. This contradiction shows that $C$ lies on no diameter other than $AC$ .