Euler Line

In any non-equilateral triangle the orthocenter (H), the centroid (G) and the circumcenter (O) are aligned. The line that contains these three points is called the Euler line.

In an equilateral triangle all three centers meet in a single point, so there’s no Euler line.

Distances between centers:

It is true that the distance from the orthocenter (H) to the centroid (G) is twice that of the centroid (G) to the circumcenter (O). Or put another way, the HG segment is twice the GO segment:

When the triangle is equilateral, the centroid, orthocenter, circumcenter, and incenter coincide in the same interior point, which is at the same distance from the three vertices.

This distance to the three vertices of an equilateral triangle is equal to from one side and, therefore, to the vertex, being h its altitude (or height).

The incenter (I) lies on the Euler line only for an isosceles triangle. In an isosceles triangle, the Euler line coincides with its axis of symmetry, which is located along the perpendicular bisector of its base (See figure above).

Therefore, when the triangle is an isosceles all four points of concurrency are collinear on the Euler line: the orthocenter (H), centroid (G), circumcenter (O), and also the incenter (I).

In an equilateral triangle, as we see in the figure above, all four triangle centers (labeled as H, I, G and O) fall into a single point.

In fact, we can assert that a triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide.

Euler’s Theorem: Distance between Incenter and Circumcenter of a triangle

Can we calculate the distance between these two centers of a triangle?

Remember that the incenter (I) is the center of the incircle, which is the largest circle that will fit inside the triangle. The incircle’s radius is called inradius (r). While, the circumcenter (O) is the center of the circumscribed circle, or circumcircle, whose circumradius (R) is equal to the distance between the circumcenter and any of the three vertices of the triangle.

So, we can calculate the distance between incenter (I) and circumcenter (O) using Euler’s Theorem, which states that the distance between the incenter and circumcenter of a triangle can be calculated by the equation:

Where OI is the distance between both centers, and R and r are the length of circumradius and inradius respectively.

See the picture below:

Orthocenter of a triangle

In a triangle ABC the orthocenter H is the intersection point of the three altitudes of the triangle.

Every triangle has three altitudes (or heights) and three sides (or bases).

An altitude of a triangle (h_a, h_b y h_c) is a perpendicular line segment from a vertex to the opposite side. This line containing the opposite side is called the extended base of the altitude.

Altitude can also be understood as the distance between the base and the vertex.

Where is the Orthocenter of a Triangle Located?

• If it’s an obtuse triangle the orthocenter is located outside the triangle (as we see in the picture above).
• If it’s an acute triangle the orthocenter is located inside the triangle.
• If it’s a right triangle the orthocenter lies on the vertex of the right angle.

Centroid of a triangle

The centroid of a triangle (or barycenter of a triangle) G is the point where the three medians of the triangle meet.

The medians of a triangle are the line segments created by joining one vertex to the midpoint of the opposite side. Since every triangle has three sides and three angles, it has three medians (m_a, m_b and m_c).

Centroid theorem: the distance between the centroid and its corresponding vertex is twice the distance between the barycenter and the midpoint of the opposite side. That is, the distance from the centroid to each vertex is 2/3 the length of each median. This is true for every triangle.

In physics, the centroid of a triangle (G) would be its center of gravity.

The centroid is always inside the triangle.

Circumcenter of a triangle

The circumcenter of a triangle (O) is the point where the three perpendicular bisectors (M_a, M_b y M_c) of the sides of the triangle intersect. It can be also defined as one of a triangle’s points of concurrency.

The perpendicular bisector of a triangle is a line perpendicular to the side that passes through its midpoint.

The circumcenter (O) is the central point that forms the origin of the circumcircle (circumscribed circle) in which all three vertices of the triangle lie on the circle.

It’s possible to find the radius (R) of the circumcircle if we know the three sides and the semiperimeter of the triangle.

The radius of the circumcircle is also called the triangle’s circumradius.

The formula for the circumradius is:

Where is the Circumcenter of a Triangle Located?

• If it’s an obtuse triangle the circumcenter is located outside the triangle (as we see in the picture above).
  
• If it’s an acute triangle the circumcenter is located inside the triangle.
• If it’s a right triangle the circumcenter lies on the midpoint of the hypotenuse (the longest side of the triangle, that is opposite to the right angle (90°). We can see an example in the figure below.
  

See the Thales’ Theorem.