Knowing two points on a circle does not define a unique circle, so it is impossible to find the centre of the circle as there are infinitely many centres possible.
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Knowledge of the coordinates of two pints on a circle is not sufficient to locate its centre.
That point is known as the "center of the circle". Every circle has one.
It is the diameter.
Incenter (apex)
A circle is classified as a two-dimensional geometric shape that is defined as the set of all points in a plane that are equidistant from a fixed point called the center. It is a type of conic section with a constant radius. The distance from the center to any point on the circle is known as the radius, and the longest distance across the circle, passing through the center, is called the diameter.
If a line segment touches two points on the circle it is called as: 1. secant; if the line extends outside the circle. 2. chord; if the line is present only within the circle. Diameter is a line which satisfies the second condition hence it is a chord. The additional condition for a chord to be known as diameter is that the chord should pass through the center of the circle!