Orthocenter, Centroid, Circumcenter and Incenter of a Triangle

Orthocenter

The orthocenter of a triangle is defined as:

The point of intersection of the three heights of a triangle

Or

The point where all the three altitudes of the triangle meet or intersect each other. 

An altitude or height is each of the perpendicular lines drawn from one vertex to the opposite side (or its extension).

See the following triangle. Point H is the orthocenter of this triangle because it is the point where all the three altitudes of the triangle are intersecting each other.

The orthocenter is different for various triangles such as isosceles, scalene, equilateral, and acute, etc. For an equilateral triangle, the centroid will be the orthocenter. In the case of other types of triangles, the position of the point where all the three altitudes intersect will vary.

• The orthocenter of an acute triangle lies inside the triangle
• The orthocenter of an obtuse triangle lies outside the triangle
• The orthocenter of a right-angled triangle lies on the vertex of the right angle

Centroid

The centroid is defined as:

The point of intersection of the three medians.

A median refers to the straight line that joins the midpoint of a side with the opposite vertex.

A centroid has the following properties:

• A centroid refers to the center of an object and it is the center of gravity
• It always lies inside the triangle
• It is the point of intersection or concurrency of three medians of the triangle

A centroid of the triangle is shown in the figure below:

The centroid divides each median into two segments, the segment joining the centroid to the vertex is twice the length of the line segment joining the midpoint to the opposite side.  In the case of  the above triangle, we can denote it as:

BG = 2GA

Hence, we can conclude that a centroid segregates the median of the triangle in the ratio of 2 : 1. We can find the centroid of a triangle by calculating the average of x- coordinate points and y - coordinate points of all three vertices of a triangle.

For instance, consider the following example:

Find the centroid of a triangle whose vertices are (3, 4), (5, 7), and (4, 7).

To find the centroid, we need to take the average of x-coordinates and y - coordinates separately.

Average of x - coordinates =

Average of Y - coordinates =

Hence, the centroid of the triangle is (4, 6)

 

Circumcenter

The circumcenter of the triangle is defined as:

The point of intersection of the three perpendicular bisectors.

A perpendicular bisector of a triangle is each line drawn perpendicularly from its midpoint. The circumcenter is the center of a triangle's circumcircle (circumscribed circle). A circumcenter of the triangle is shown in the figure below:

Incenter

The incenter of a triangle is defined as:

The point of intersection of the three angle bisectors

The angle bisectors of a triangle are each one of the lines that divide an angle into two equal angles. The incenter is the center of the circle inscribed in the triangle.

Thus, we can say that the incenter of a triangle is the intersection point of all the three angle bisectors of a triangle. The incenter is always equidistant from the three sides of the triangle.

The incenter of a triangle is shown in the figure below:

Line of Euler

The orthocenter, the centroid, and the circumcenter of a non-equilateral triangle are aligned. It means that they lie on the same straight line, called a line of Euler.

You can see in the below figure that the orthocenter, centroid and circumcenter all are lying on the same straight line and are represented by O, G, and H. This straight line is the line of Euler.