## what is the equidistant from the sides of a triangle?

### incenter

The equidistant from the sides of a triangle refers to the set of points that are at an equal distance from each of the three sides of the triangle. These points form a line called the perpendicular bisector of each side.

To understand this concept, let’s take a triangle ABC as an example. The perpendicular bisector of side AB is the line that is equidistant from points A and B. Similarly, the perpendicular bisector of side BC is the line that is equidistant from points B and C, and the perpendicular bisector of side AC is equidistant from points A and C.

The equidistant points from all three sides of the triangle will lie at the intersection of these three perpendicular bisectors. This point is called the circumcenter of the triangle, denoted as O.

The circumcenter is an important point in a triangle because it is the center of the circumcircle, which is a circle that passes through all three vertices of the triangle. The radius of the circumcircle is the distance from the circumcenter (O) to any vertex of the triangle (A, B, or C).

It’s also worth noting that the circumcenter lies inside the triangle for an acute triangle, outside the triangle for an obtuse triangle, and coincides with the midpoint of the hypotenuse for a right triangle.

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