lines =
 {{Line[{{243.8`, 77.`}, {467.4`, 12.`}}], Line[{{356.8`, 32.`}, {363.2`, 120.`}}]},
  {Line[{{291.8`, 130.`}, {476.`, 210.5`}}], Line[{{346.`, 245.`}, {393.8`, 158.`}}]},
  {Line[{{103.2`, 327.`}, {245.2`, 110.5`}}], Line[{{163.8`, 211.5`}, {230.2`, 250.`}}]},
  {Line[{{47.4`, 343.`}, {87.4`, 108.5`}}], Line[{{54.6`, 225.`}, {139.6`, 220.`}}]},
  {Line[{{371.`, 506.5`}, {384.6`, 277.`}}], Line[{{366.`, 394.5`}, {451.8`, 372.`}}]},
  {Line[{{264.6`, 525.5`}, {353.8`, 294.5`}}], Line[{{241.`, 398.`}, {321.`, 411.5`}}]},
  {Line[{{113.2`, 484.5`}, {296.`, 304.5`}}], Line[{{163.2`, 347.`}, {213.2`, 406.5`}}]},
  {Line[{{459.6`, 604.5`}, {320.2`, 466.5`}}], Line[{{332.4`, 596.5`}, {402.4`, 528.5`}}]},
  {Line[{{288.2`, 630.5`}, {199.6`, 446.5`}}], Line[{{176.`, 585.5`}, {256.`, 530.5`}}]},
  {Line[{{138.8`, 615.5`}, {81.8`, 410.`}}], Line[{{38.2`, 553.`}, {122.4`, 507.`}}]},
  {Line[{{232.4`, 795.`}, {461.8`, 727.`}}], Line[{{345.2`, 774.5`}, {345.2`, 688.`}}]},
  {Line[{{27.4`, 671.5`}, {206.8`, 763.5`}}], Line[{{104.6`, 728.`}, {161.8`, 647.`}}]}};

(points = Point /@ RegionCentroid /@ DiscretizeRegion /@ RegionUnion @@@ lines); //
   Timing // First

Graphics[{lines, {Red, PointSize@0.02, points}}, Frame -> True]

points = Cases[Show[DiscretizeRegion /@ RegionIntersection @@@ lines],
   {a_Real, b_Real} :> Point[{a, b}], Infinity];