Understanding Triangle Centers: Circumcenter, Incenter, Centroid, and Orthocenter

A polygon is the portion of the plane bounded by a closed polyline. A polygon is a plane shape with straight sides. A plane is a flat surface with no thickness.
Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is “closed” (all the lines connect up).

1. Circumcenter

The circumcenter is the center of the triangle’s circumcircle, one that passes through the three vertices of the triangle.
It is located at the intersection of the three side bisectors. As we already know the line bisector is the locus of all the circle centers that pass through the endpoints of a segment. As the three sides of a triangle are segments, if we draw the three side bisectors, we will get a point that will be the center of a circle that passes through the three vertices of the triangle, the Circumcircle.
It is called graphically with the letter O.

• In an acute triangle, the circumcircle’s center is inside the triangle.
• In an obtuse triangle, the circumcircle’s center is outside the triangle.
• In a right triangle, the circumcircle’s center is the midpoint of the hypotenuse.

2. Incenter

The incenter is the center of the triangle’s incircle, also known as inscribed circle, it is the largest circle that will fit inside the triangle. Each of the triangle’s three sides is a tangent to the circumference.
It is located at the intersection of the three angle bisectors. As we already know the angle bisector is the locus of all the circle centers that are tangent to the sides of an angle. As a triangle has three angles, if we draw the three angle bisectors, we will get a point that will be the center of a circle that is tangent to the three sides of the triangle, the Incircle.
It is called graphically with the letter I.
Incenter is always inside the triangle.

3. Centroid

The barycenter or centroid is located on the intersection of the three medians of the triangle, and is equivalent to the center of gravity of the triangle.
It is called graphically with the letter G.
The medians of the triangle are each segments which join each vertex of the triangle with the midpoint of the opposite side.
Barycenter is always inside the triangle.

4. Orthocenter

The Orthocenter is the point located at the intersection of the triangle heights.
It is called the height of a triangle the segment which join a vertex of a triangle to the opposite side -or its extension- forming a right angle (90 °). All triangles have three heights.
It is called graphically with the letter H.

• In an acute triangle, the orthocenter is inside the triangle.
• In an obtuse triangle, the orthocenter is outside the triangle.
• In a right triangle, the orthocenter is the vertex of the triangle’s right angle.