Triangle Centers and Geometric Loci Explained

Triangle Centers

Incenter

The angle bisectors of a triangle are the lines that divide the interior angles into two equal parts. The three bisectors intersect at a point equidistant from the three sides, called the incenter, which serves as the center of the inscribed circle.

Circumcenter

The perpendicular bisectors are lines perpendicular to the sides of a triangle at their midpoints. These three lines intersect at a point called the circumcenter, which is equidistant from the three vertices and acts as the center of the circumcircle. The circumcenter can be located inside or outside the triangle.

Centroid

The medians are segments joining a vertex with the midpoint of the opposite side. They intersect at a point called the centroid, which represents the center of gravity of the triangle.

Orthocenter

The altitudes of a triangle are segments drawn from a vertex perpendicular to the opposite side or its extension. The three altitudes intersect at a point called the orthocenter, which may be located inside or outside the triangle.

Geometric Loci

A locus defines a specific set of positions in a plane or space.

Perpendicular Bisector

This line is perpendicular to a segment at its midpoint. All points on this line are equidistant from the endpoints of the segment. It is defined as the locus of points equidistant from the extremes of the segment.

Angle Bisector

This is the line that divides an angle into two equal parts through its vertex. Each point on the bisector is the same distance from both sides of the angle. It is defined as the locus of points equidistant from the sides of the angle.

Circle

A circle is a closed plane curve where all points are equidistant from a fixed center point. It is the locus of points at a constant distance, equal to the radius, from a fixed center point.

Capable Arc

The capable arc is the locus of points in a plane from which the endpoints of a segment are seen under a specific angle.