construction a triangle with bisectors and one side(case1)

construction a triangle with bisectors seems very difficult. Even for Isosceles triangle. This is case I which can be constructed but with the limitation of the bisectors.

construction of a triangle with two side s and inscribe circle

 I have a case which is to construction a triangle with two sides and a inscribe circle.

the case make me confused as it seems a mission impossible. I have plot the graphic of the radius and inner center. but it seems I can only know there is the limitation of the radius and possible two solutions for the construction.

 I had try the simple case which is isosceles triangle but I am also failed for general case. And I got a very interesting and surprising fact is that for the biggest inscribe circle, there is gold ratio point. the angle and tangent point and inner center are all at golden ratio points.

The last picture is the process to construction this triangle with biggest inscribe circle. another interesting fact is also showed in the picture.

I suspect that there is no general compass and straight method to construction for this case.