Four dogs chasing each other from the corners of a square

Question

Four dogs sit at the four corners of a square field of side 100 metres. At the same instant they all start running at the same constant speed, each one always heading directly towards the next dog clockwise around the square. Their paths curl inwards and they meet at the centre.

How far does each dog run before they meet? Do not try to find the shape of the paths; there is a shortcut.

Accepted Answer

Each dog runs exactly 100 metres, the side of the square. The symmetry. At every moment the situation is symmetric under a quarter turn about the centre, so the four dogs always sit at the corners of a square (smaller and rotated as time passes). In particular, dog A always runs straight at dog B, and dog B is always running at right angles to the line from A to B (it is running along the next side of the current square). The closing speed. The distance from A to B shrinks because A runs directly towards B at speed v. B's own motion is perpendicular to the line AB, so it neither increases nor…

Four dogs chasing each other from the corners of a square

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