Question

Three points are located at the vertices of an equilateral triangle whose side equals $a$. They all start moving simultaneously with velocity $v$ constant in modulus, with the first point heading continually for the second, the second for the third, and the third for the first. The time at which the points converge is given as $t=\frac{2a}{xv}$. Find $x$.

Answer 1

From the symmetry of the problem all the three points are always located at the vertices of equilateral triangles of varying side length and finally meet at the centroid of the initial equilateral triangle whose side length is $a$, in the sought time interval (say $t$).
Let us consider an arbitrary equilateral triangle of edge length $l$ (say).
Then the rate by which 1 approaches 2, 2 approaches 3, and 3 approaches 1, becomes:
$\frac{-dl}{dt}=v-v\cos \left(\frac{2\pi }{3}\right)$
On integrating: $-{\int }_a^{0}dl=\frac{3v}{2}{\int }_0^{t}dt$
$a=\frac{3}{2}vt$ so $t=\frac{2a}{3v}$