Incenter, Orthocenter, Circumcenter, Centroid. Date: 01/05/97 at From: Kristy Beck Subject: Euler line I have been having trouble finding the Euler line. Orthocenter: Where the triangle’s three altitudes intersect. Unlike the centroid, incenter, and circumcenter — all of which are located at an interesting point of. They are the Incenter, Orthocenter, Centroid and Circumcenter. The Incenter is the point of concurrency of the angle bisectors. It is also the center of the largest.

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Here are the 4 most popular ones: Also, construct the altitude DM. It is pictured below as the red dashed line. Triangle Centers Where is the center of a triangle? The orthocenter H of a triangle is the point of intersection of the circumcenterr altitudes of the triangle.

Thus, the circumcenter is the point that forms the origin of a circle in which all three vertices of the triangle lie on the circle. It should be noted that the circumcenter, in different cases, may lie outside the triangle; in these cases, the perpendicular bisectors extend beyond the sides of the triangle.

Orthocenter, Centroid, Circumcenter and Incenter of a Triangle

Hide Ads About Ads. A median is a segment constructed from a vertex to the midpoint of the subtending side of the triangle.


The centroid is the point of intersection of the three medians. Note that sometimes the edges of the triangle have to be extended outside the triangle to draw the altitudes. A perpendicular bisectors of a triangle is each line drawn perpendicularly from its midpoint. Centroid, Circumcenter, Incenter and Orthocenter For each of those, the “center” is where special lines cross, so it all depends on those lines!

In a right triangle, the orthocenter falls on a vertex of the triangle. Orthocenter Draw a line called the “altitude” at right angles to a side and going through the opposite corner.

Where all three lines intersect is the “orthocenter”:. So, label the point of intersection H’. To see that the incenter is in fact always inside the triangle, let’s take a look at an obtuse triangle and a right triangle. The circumcenter is the center of the circle such that all three vertices of the circle are the same distance away from the circumcenter.

Triangle Centers

The orthocenterthe centroid and the circumcentef of a non-equilateral triangle are aligned ; that is to say, they belong to the same straight line, called line of Euler. The centroid G of a triangle is the point of intersection of the three medians of the triangle. Thus, GH’and C are collinear. Where all three lines intersect is the centroidwhich is also the “center circumcentdr mass”: Then the orthocenter is also outside the triangle. Defining them first is necessary in order to see their relationship with each other.


Let’s look at each cfntroid The centroid of a triangle is constructed by taking any given triangle and connecting the midpoints of each leg of the triangle to the opposite vertex.

Where all three lines intersect is the “orthocenter”: Circumcenter Draw a line called a “perpendicular bisector” at right angles to the midpoint of each side.

The Centroid, Circumcenter, and Orthocenter Are Collinear – Wolfram Demonstrations Project

The line segment created by connecting these points is called the median. Where all three lines intersect is the centroidwhich is also the “center of mass”:.

The incenter is the center of the circle inscribed in the triangle. Since H is the orthocenter, H is on DM by the definition of orthocenter.


There is an interesting relationship between the centroid, orthocenter, and circumcenter of a triangle. In fact, it can be outside the triangle, as in the case of an obtuse triangle, or it can fall at the midpoint of the hypotenuse of a right triangle. Thus, H’ is the orthocenter because it is lies on all three altitudes.

An altitude is a line constructed from a vertex to the subtending side of the triangle and is perpendicular to that side.