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.

Nice problems, exploiting some of the nicer properties of incircles
and equilateral triangles....

1. The point is obviously the incenter, with radius = A/s = 10.
        Thus, coordinates are (10,10).

2. The new plot is an equilateral triangle; so sum of the distances 
        from P, the common internal point, to each of the sides is 
        always equal to the side of the triangle (= 100).  Also, the 
        ratio of the areas is the same as the ratio of the corresponding 
        distances from P to each of the 3 sides, in proportion 5:4:3.  

        Draw the equilateral triangle XYP, with base XY on the x-axis and 
        vertex at point P.  X,Y divide the side AB into segments AX:XY:YB = 4:3:5
        (the altitudes of the equilateral triangles with sides AX,XY,YB are
        exactly the same as the distances from P to sides b,c,a!).
        Drop altitude from P onto XY at Z.  It follows that AZ:ZB = 5.5:6.5.
        Since AB=100, this makes the x coordinate of Z (and P) = 100*(-1/24).
        y coordinate of P is PZ = sqrt(3)/2 * XY = sqrt(3)/2 * 100/4.

        So, coordinates are (-25/6, 25*sqrt(3)/2).