3-points in the 3-4-5 triangle

The problem was set in Gerald Brown's post to geometry-puzzles 25.10.01.

Here is this post:

   A 3-point of a triangle is defined as a point where there exists
three distinct lines through the point and each line divides the
triangle into two polygons of equal area. Clearly, the centroid of the
triangle is a 3-point of the triangle. What percent of the area of a
triangle is the area of the set of 3-points of the triangle?

As mere illustration of the problem, I present here an example of 3-points in the 3-4-5 triangle.

Here coordinates of the verticies of the triangle are 
C{0,0}, B{3,0}, A{0,4},
such that a = CB =3, b = AC = 4, and c = AB = 5.
AD, BE, and CF are medians, and G is the centroid (as cross-point of medians). 
Each of three red lines divides the area triangle to two equal parts.
Note that each red line is parallel to the
relevant side of the triangle ABC and is at a distance 
of (1-1/2^(1/2)) of the length of an altitude to that side.  

Each pair of red lines meet at point 
which is on on of the medians. 
This is in full conformity with  Walter Whiteley assertion.

The coordinates of the 3-points in this simple case are:

{3  (1 - 1/2 ^ (1/2 )), 4  (1 - 1/2 ^ (1/2 ))}, 
{3  (1 - 1/2 ^ (1/2 )), 4  (2 ^ (1/2 ) - 1 )}, 
{3  (2 ^ (1/2 ) - 1 ),  4  (1 - 1/2 ^ (1/2 ))}.

The relative area covered by 3-points is given here.