Dividing a Triangle

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.

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.
Solution
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