Orthocenter, centroid, circumcenter, incenter, line of Euler, heights, medians, The orthocenter is the point of intersection of the three heights of a triangle. Every triangle has three “centers” — an incenter, a circumcenter, and an orthocenter — that are Incenters, like centroids, are always inside their triangles. Triangles have amazing properties! Among these is that the angle bisectors, segment perpendicular bisectors, medians and altitudes all meet with the .

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Centroid, Circumcenter, Incenter and Orthocenter For each of those, the “center” is where special lines cross, so it all depends on those lines!

The angle bisectors of a triangle are each one of the lines that divide an angle into two equal angles. The circumcenter is the point of intersection of the three perpendicular bisectors.

### Math Forum – Ask Dr. Math

The orthocenter is the point of intersection of the three heights of anc triangle. Check out the cases of the obtuse and right triangles below. It is the point forming the origin of a circle inscribed inside the 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.

Circumcenterconcurrency of the three perpendicular bisectors Incenterconcurrency of the three angle bisectors Orthocenterconcurrency of the three altitudes Centroidconcurrency of the three medians. You see that even though the circumcenter is outside the triangle in the case of the obtuse triangle, it is still equidistant from all three vertices of the triangle.

### Triangle Centers

Circumcenterconcurrency of the three perpendicular bisectors. The circumcenter is not always inside the triangle.

The incenter is rothocenter point of intersection of the three angle bisectors. Where all three lines intersect is the “orthocenter”:. The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side.

Note that sometimes the edges of the triangle have to be extended outside the triangle to draw the altitudes. There are actually thousands of centers!

Draw a line called the “altitude” at right angles to a side and going through the opposite corner.

The centroid is the point of intersection of the three medians. Summary of geometrical theorems summarizes the proofs of concurrency of the lines that determine these centers, as well as many other proofs in geometry. Where all three lines intersect is the centroidwhich is also the “center of mass”: Where all three lines intersect is the center of a triangle’s adn, called the “circumcenter”: If you have Geometer’s Inceenter and would like to see the GSP construction of the orthocenter, click here to download it.

Incenter 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 a triangle’s “incircle”, called the “incenter”: This page summarizes some of them. The incenter is the center of the circle inscribed in the triangle. Barycentric Coordinateswhich provide a way of calculating these triangle centers see each of the triangle center pages for the barycentric coordinates of that center.

It is pictured below as the red dashed line. Circumcenterconcurrency of the three perpendicular bisectors Incenterconcurrency of the three angle bisectors Orthocenterconcurrency of the three altitudes Centroidconcurrency of the three medians For any triangle all three medians intersect at one point, known as the centroid.

Draw a line called a “perpendicular bisector” at right angles to the midpoint of each side. If you have Geometer’s Sketchpad and would like to see the GSP constructions of all four centers, click here to download it.

## Orthocenter, Centroid, Circumcenter and Incenter of a Triangle

If you have Geometer’s Sketchpad and would like to see the GSP construction of the incenter, click here to download it. Orthcoenter different kinds of “centers” of a triangle can be found.

A height is each of the perpendicular lines drawn from one vertex to the opposite side or its extension. It can be used to generate all of the pictures above. Where all three lines intersect is the circumcenter.