trajectories[{pt1_, pt2_, pt3_}] := NDSolveValue[{
{x1'[t], y1'[t]} == -Normalize[{x2[t], y2[t]} - {x1[t], y1[t]}],
{x2'[t], y2'[t]} == -Normalize[{x3[t], y3[t]} - {x2[t], y2[t]}],
{x3'[t], y3'[t]} == -Normalize[{x1[t], y1[t]} - {x3[t], y3[t]}],
{x1[0], y1[0]} == pt1,
{x2[0], y2[0]} == pt2,
{x3[0], y3[0]} == pt3
}, {x1, y1, x2, y2, x3, y3}, {t, 0, 1}]
pos = Partition[
Through[trajectories[{{0.2916, 0.2916}, {0.7083, 0.2916}, {0.5,
0.5}}][t]], 2];
Graphics[{
Red, Line@Transpose[Table[pos, {t, 0, 1, 0.05}]],