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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Barycentric Coordinateswhich provide a way of calculating these triangle centers see each of the triangle center pages for the barycentric coordinates of that center. 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.

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.

## Triangle Centers

Contents of this section: Draw centroif 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”:. Note that sometimes the edges of the triangle have to be extended outside the triangle to draw the altitudes.

Where all three lines intersect is the “orthocenter”: Like the circumcenter, the orthocenter does not have to be inside the triangle. Draw a line called the “altitude” at right angles to a side and going through the opposite corner. Orthocenterconcurrency of the three altitudes.

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

The centroid is the point of intersection 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 amd three vertices of the triangle.

Circumcenterconcurrency of the three perpendicular bisectors Incenterconcurrency of the three angle bisectors Orthocenterconcurrency of the three altitudes Centroidconcurrency of the three medians.

Circumcenterconcurrency of the three perpendicular bisectors.

Circumcenter Draw a line called a “perpendicular bisector” at right angles to the midpoint of each side. Draw a line called a “perpendicular bisector” at right angles to the midpoint of each side.

You can look at the above example of an acute triangle, or the below examples of an obtuse triangle and a right triangle to see that this is the case. Like the centroid, the incenter is always inside the triangle. The three altitudes lines perpendicular to one side that pass through the remaining vertex of the triangle intersect at one point, known as the orthocenter of the triangle.

In this assignment, we will be investigating 4 different triangle centers: If you have Geometer’s Sketchpad and would like to see the GSP constructions of all four centers, 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”: If you have Geometer’s Sketchpad and would like to see the GSP construction of the circumcenter, click here to download it.

### Math Forum – Ask Dr. Math

Circumcenterconcurrency of the three perpendicular bisectors Incenterconcurrency of orhtocenter 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. The circumcenter is not always inside the triangle. Triangle Centers Where is the center of a triangle?

Orthocenter Draw a line called the “altitude” at right angles to a side and going through the opposite corner. Thus, the radius of the circle is the distance between the circumcenter and any of the triangle’s three vertices. See the pictures below for examples of this. You see the three medians as the dashed lines in the figure below.

It is pictured below as the red dashed line. Orthocenter, Centroid, Circumcenter and Incenter of a Triangle. There are actually thousands of centers!

The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side.

### Triangle Centers

Hide Ads About Ads. The incenter is the last triangle center we will be investigating. Centroidconcurrency of the three medians. The circumcenter is the center of a triangle’s circumcircle circumscribed circle.

There is an interesting relationship between the centroid, orthocenter, and circumcenter of a triangle. The line segment created by connecting these points is called the median. The centroid is the center of a triangle that can be thought of as the qnd of mass. Where all three lines intersect is the “orthocenter”:.