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Home > Science > What is the Difference Between Circumcenter, Incenter, Orthocenter and Centroid?

Guilherme Mazui

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What is the Difference Between Circumcenter, Incenter, Orthocenter and Centroid?

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The difference between circumcenter, incenter, orthocenter, and centroid lies in their definitions and the properties they possess:

  1. Circumcenter (O): The circumcenter is the point that is equidistant from all the vertices of the triangle. It is the center of the circumcircle, which is a circle passing through all three vertices of the triangle.
  2. Incenter (I): The incenter is the point that is equidistant from the sides of the triangle. It is the point of intersection of the three angle bisectors of the triangle.
  3. Orthocenter (H): The orthocenter is the point where all the altitudes of the triangle intersect. The altitudes are the line segments connecting each vertex to the midpoint of the opposite side.
  4. Centroid (G): The centroid is the point of intersection of the medians of the triangle. The medians are the line segments connecting each vertex to the midpoint of the opposite side. The centroid divides each median in a 1:2 ratio, and the center of mass of a uniform, triangular lamina lies at this point.

In summary, the circumcenter is associated with the vertices, the incenter with the sides, the orthocenter with the altitudes, and the centroid with the medians of the triangle. These points are used to analyze and solve various geometric problems in triangles.

On this page
  • What is the Difference Between Circumcenter, Incenter, Orthocenter and Centroid?
  • Comparative Table: Circumcenter, Incenter, Orthocenter vs Centroid

    • Comparative Table: Circumcenter, Incenter, Orthocenter vs Centroid

    Here is a table comparing the differences between the circumcenter, incenter, orthocenter, and centroid of a triangle:

    Property Circumcenter Incenter Orthocenter Centroid
    Definition The point where the perpendicular bisectors of a triangle meet. The point where the angle bisectors of a triangle meet. The point where the medians of a triangle meet. The intersection point of all three angle bisectors of a triangle.
    Associated Circle Circumcircle, which is a circle passing through all three vertices of the triangle. Incircle, which is a circle inscribed in the triangle that is tangent to all three sides. No associated circle. No associated circle.
    Relation to Side Lengths Equidistant from the three sides of the triangle. Always inside the triangle. Position varies depending on the type of triangle (e.g., inside, outside, or on the vertex of a right-angled triangle). Divides each median in a 1:2 ratio.
    Construction Method Create any two perpendicular bisectors to the sides of the triangle. The point of intersection gives the circumcenter. Create any two angle bisectors to the sides of the triangle. The point of intersection gives the incenter. Create any two medians of the triangle. The point of intersection gives the centroid. Create any two altitudes of the triangle. The point of intersection gives the orthocenter.

    Please note that the orthocenter, centroid, and circumcenter of a non-equilateral triangle are aligned and lie on the same straight line, called the line of Euler.

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    Guilherme Mazui

    Guilherme Mazui is graduated in journalism from the Federal University of Minas Gerais (UFMG) and a master's degree in Communication from the University of São Paulo (USP). In addition, he has experience in advertising writing and has worked as a content editor in several companies.