Circumcenter, Orthocenter, Incenter, and Centroid

The circumcenter, the orthocenter, the incenter, and the centroid are points that represent the intersections of different internal segments of a triangle. For example, we can obtain intersection points of perpendicular bisectors, bisectors, heights and medians.

In this article, we will explore the circumcenter, orthocenter, incenter, and centroid of a triangle. We will look at their diagrams and will learn about some of their main characteristics.

GEOMETRY
orthocenter of a triangle

Relevant for

Learning about the circumcenter, orthocenter, incenter, and centroid.

See definitions

GEOMETRY
orthocenter of a triangle

Relevant for

Learning about the circumcenter, orthocenter, incenter, and centroid.

See definitions

Circumcenter of a triangle

The circumcenter of a triangle represents the point of intersection of the perpendicular bisectors of the three sides of the triangle. The following is the diagram of the circumcenter.

circumcenter of a triangle

Remember that the perpendicular bisectors are the perpendicular segments that start from the midpoint of a segment. In this case, the perpendicular bisectors pass through the midpoints of all three sides of the triangle.

Alternatively, we can also define the circumcenter as the center of the circumscribed circle. Let us remember that a circumscribed circle is the circle that passes through all the vertices of a polygon, as we see in the following diagram.

circumcenter of a triangle with a circumscribed circle

The location of the circumcenter is different depending on the type of triangle:

  • In an acute triangle, the circumcenter is always located inside the triangle.
  • In an obtuse triangle, the circumcenter is located outside the triangle.
  • In a right triangle, the circumcenter is located on the hypotenuse of the triangle.
  • In an equilateral triangle, the circumcenter is located in the same position as the centroid, incenter, and orthocenter.

If you want to learn more about the circumcenter of a triangle, check out our article about the Circumcenter of a triangle.


Orthocenter of a triangle

The orthocenter of a triangle represents the point of intersection of the three heights of the triangle. We can observe the orthocenter in the following diagram.

orthocenter of a triangle

Remember that the heights of the triangle are the lines that are perpendicular to the sides and that join a vertex with the opposite side. That is, the heights form 90° angles with their corresponding side.

The location of the orthocenter varies depending on the type of triangle:

  • In acute triangles, the orthocenter is located inside the triangle.
  • In obtuse triangles, the orthocenter is located outside the triangle.
  • In right triangles, the orthocenter is located at the vertex opposite the hypotenuse.
  • In equilateral triangles, the orthocenter is in the same position as the centroid, incenter, and circumcenter.

If you want to learn more about the orthocenter of a triangle, check out our article about the Orthocenter of a triangle.


Incenter of a triangle

The incenter of a triangle represents the point of intersection of the bisectors of the three interior angles of the triangle. The following is a diagram of the incenter of a triangle:

incenter of a triangle

Remember that the bisectors are the line segments that divide the angles into two equal parts.

Alternatively, the incenter of a triangle can also be defined as the center of a circle inscribed in the triangle. Also, an inscribed circle is the largest circle that fits inside the triangle.

incenter of a triangle with an incircle

The incenter is always located inside the triangle, no matter what type of triangle we have. However, as we already mentioned, the incenter of equilateral triangles is in the same position as the incenter, the orthocenter, the circumcenter, and the centroid.

You can learn more about the incenter of a triangle in our article about the Incenter of a triangle.


Centroid of a triangle

The centroid of a triangle represents the point of intersection of the three medians of the triangle. The following is a diagram of the centroid in a triangle:

centroid of a triangle

Remember that the medians of the triangle are the line segments that join a vertex with the midpoint of the opposite side. We can see that each of the medians divides the triangle into two smaller congruent triangles.

The centroid is always located inside the triangle no matter what type of triangle we have. However, for equilateral triangles, the centroid, orthocenter, incenter, and circumcenter are located in the same position.

You can learn more about the centroid of a triangle in our article about the Centroid of a triangle.


See also

Interested in learning more about Platonic solids and geometric figures? Look at these pages:

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