The locus of points from which the angular sizes
            of two legs of the 3-4-5 triangle are the same.
Coordinates of the vertices of the triangle are: C{0,0}, B{3,0}, A{4,0}.
The red line is the cubic:

      = 0,

or in the parametric form:

    y = ,   x = ,

The point with angle = (Napoleon Point),
and the foot of altitude from C to hypotenuse are shown.
For Napoleon Point,  k = , and coordinates are:
     .
The point with angle = corresponds to    k = , and has coordinates
     .
     
     The limit of curve at point {3,0} corresponds to angle ArcTan[4/3] = 1301,
     and  k = -.
     The dashed blue right line  is the limit of the red curve near point {0,0}: y = x.
     

Note that the shown red curve is only the part of the full locus!!