Center
A center is a point that is the same distance from all the extremities of a figure.
1. Center of a circle
The center of a circle is the point which is equidistant from all points on the circle. In the figure below, C is the center.
The center point is often used to label the whole circle. The figure below would be called "the circle C".
For more on this see Circle definition.
2. Center of a regular polygon
The center of a regular polygon is the point that is same distance from all its
vertices.
For more on this see Center of a regular polygon.
2. Centers of triangles
Triangles have various different centers depending on how they are derived.
Below, for example, is the incenter  the point where the three angle bisectors meet.
For an overview of four common triangle centers see
Overview of triangle centers.
Definition of the center of a polygon
How to find the center of a circle with compass and straightedge or ruler. This method relies on the fact that, for any chord of a circle, the perpendicular bisector of the chord always passes through the center of the circle. By applying this twice to two different chords, the center is established where the two bisectors intersect. A Euclidean construction
How find the center of a circle using any rightangled object. This method works as a result of Thales Theorem. The diameter of a circle subtends a right angle to any point on the circle.
Points of concurrency  the point where three or more lines intersect
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