It will always lie on a diameter.
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It is the diameter of the circle.
Yes. Any triangle can be inscribed within a circle, although the center of the circle may not necessarily lie within the triangle.
Assuming all the vertices of the segmentation lie on the circle, then you can choose any three of them as the corners of a triangle circumscribed by the circle. The perpendicular bisectors of the sides of that triangle intersect at the center of the circle.
If the sides of the triangle are equal in length to the radius of the circle, then you can simply place the two ends of the hypotenuse on the perimeter of the circle, and it's remaining corner will lie at the circle's center.
No it's made by the circle that how they get the circle not the center.