Particle on vertex of a polygon moving towards adjacent particle.

geometry algebra-precalculus integration classical-mechanics vectors

Suppose we have a regular polygon with $n$ sides. On each vertex, there is a particle. Every particle moves in such a way that its velocity vector $(\vec{v})$ always points towards particle next to it. The velocity vectors of all particles are equal in magnitude.
How do we show that the meet at centroid of the polygon.

enter image description here

Actually, the problem was in physics, and it asked the time particle took to reach the centroid. The time can easily be calculated using resolution of components of vectors.


Based on simple physics the velocity of the particle may be broken into two components - one toward the center and one perpendicular to the line connecting a particle with the center. As all particle has the same velocity, all move in tandem along congruent curves. Since there is always a component to the center, equal for all, they will meet at the center after a finite time that could be calculated... do you know how to follow from here?

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