circle inscribe a triangle

[shahar]

Why there only one cycle and only one inscribe a triangle? Why not two or three?

 

[Dr.Peterson]

This doesn't make grammatical sense.

Are you asking why a triangle has only one incircle (inscribed circle)?

Why would it have more than one? Are you aware of how to construct the incircle? That construction produces only the one circle.

 

[pka]

Why there only one cycle and only one inscribe a triangle? Why not two or three?

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.

 

[shahar]

OK.
The next question (of mine) is:
Can I inscribe a triangle by one ellipse? And only one? Why not two or more?

 

[Dr.Peterson]

OK.
The next question (of mine) is:
Can I inscribe a triangle by one ellipse? And only one? Why not two or more?

Ellipses have more variety, being described by two "radii", not just one. So I would expect many different ellipses to be inscribed in the same triangle (and maybe even more than one with a given shape).