You are asking about the incircle if a triangle. The centre of that circle is called the incenter which is the one point at which all three angular bisectors meet. Hence the one incentre is equally from all three sides of the triangle.Why there only one cycle and only one inscribe a triangle? Why not two or three?
OK.
The next question (of mine) is:
Can I inscribe a triangle by one ellipse? And only one? Why not two or more?