Dividing a Triangle

  1. Al, Bill and Carl contributed $50, $40 and $30 respectively toward the purchase of a triangular piece of ground. They'd like to divide the land into three smaller triangles, in the same proportions, such that the ratios of the areas are 5:4:3. The vertices of the triangle lie at (0,0), (30,0), and (0,40) [all lengths in meters.] Al wants to own the longest side, Bill the second longest, and Carl the shortest. Their three triangles meet at a shared central point. Find the coordinates of that point.
  2. The three were so happy at their last purchase, they bought another plot of land, wanting to divide it in the same ratio of areas. The vertices lie at (-50,0), (50,0), and (0,50sqrt3). If Al is to own the side between the last two points, and Carl is to own the side between the first two points, find the coordinates of the internal point again.

Source: Original.

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