Divide a unit circle into \(n\) equal sector each of arc length \(\cfrac{1}{n}\), \(n\) runners occupy these \(n\) sections. On average, one runner is in each section, and so, their average and possible consecutive distance for two consecutive runners along the path is \(\cfrac{1}{n}\).
This is true for all numbers of runners greater than \(3\), with constant distinct speeds.
And the lonely runner conjecture is proved. No pigeonholes needed.
