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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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Construct a line through points C and G so that it intersects DM. If you have Geometer’s Sketchpad and would like to see the GSP constructions of all four centers, click here to download it.

The Centroid, Circumcenter, and Orthocenter Are Collinear

It should be noted that the orthocenter, in different cases, may lie outside the triangle; in these cases, the altitudes extend beyond the sides of the triangle. The circumcenter is the center of a triangle’s circumcircle circumscribed circle. Remember, the altitudes of a triangle do not go through the midpoints of the legs unless you have a special triangle, like an equilateral triangle.

Draw a line called the “angle bisector ” from a corner so that it splits the angle in half Where all three lines intersect is the center of circumcennter triangle’s “incircle”, called the “incenter”:.

The orthocenterthe centroid and the circumcenter of a non-equilateral triangle are aligned ; that is centriod say, they belong to the same straight line, called line of Euler. The incenter is the point of intersection of the three angle bisectors. The radius of the circle is obtained by dropping a perpendicular from the incenter to any of the triangle legs.


An altitude of the triangle is circmcenter called the height. Thus, GH’and C are collinear. Centroid Draw a line called a incenetr from a corner to the midpoint of the opposite side.

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The orthocenter is the point of intersection of the three heights of a triangle. Where all three lines intersect is the “orthocenter”: The centroid divides each median into two segmentsthe segment joining the centroid to the vertex is twice the length of the length of the line segment joining the midpoint to the opposite side. For each of those, the “center” is where special lines cross, so it all depends on those lines!

Check out the cases of the obtuse and right triangles below. There is an interesting relationship between the centroid, orthocenter, and circumcenter of a triangle. It clrcumcenter the balancing point to use if you want to balance a triangle on the tip of a pencil, for example.

Where all three lines intersect is the “orthocenter”:.

Math Forum – Ask Dr. Math

Centroid, Circumcenter, Incenter and Orthocenter For each centgoid those, the “center” is where special lines cross, so it all depends on those lines! It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle. A perpendicular bisector is a line constructed at the midpoint of a side of a triangle at a right angle to that side.


Therefore, DM meets EF at a right angle. The line that connects the centroid Gthe orthocenter Hand the circumcenter C is called the Euler Line.

The incenter is the center of the circle inscribed in the triangle. Yet, by the given hypothesis, H is the orthocenter.

An angle bisector is a line whose points are all equidistant from the two sides of the angle. Then the orthocenter is also outside the triangle. The incenter I of a triangle is the point of intersection of the three angle bisectors of the triangle.

Thus, the incenter I is equidistant from all three sides of the triangle. Thus, the radius of the circle is the distance between the circumcenter and any of the triangle’s three vertices. The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side.

The circumcenter is not always inside imcenter triangle.

It is found by finding the midpoint of each leg of the triangle and constructing a line perpendicular to that leg at its midpoint.