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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Inscribed Circle and Angle Bisectors.

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

You see the three incentfr as the dashed lines in the figure below. It is found by finding the midpoint of each leg of the triangle and constructing a line perpendicular to that leg at its midpoint. The angle bisectors of a triangle are each one of the lines that divide an angle into two equal angles.

Like the centroid, the incenter is always inside the triangle. The incenter is the center of the circle inscribed in the triangle. 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.

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

The line segment created by connecting these points is called the median. Incenterconcurrency of the three angle bisectors. For each of those, the “center” is where special lines cross, so it all depends on those lines! The circumcenter is the point of intersection of the three perpendicular bisectors. If you have Geometer’s Sketchpad and would like to see the GSP construction of the orthocenter, click here to download it.

Orthocenterconcurrency of the three altitudes. Circumcenterconcurrency of the three perpendicular bisectors. Draw a line called the “altitude” at right angles to a side and going through the opposite corner.

If you have Geometer’s Sketchpad and would like to see the GSP centgoid of all four centers, click here to download it.

The incenter is the last triangle center we will be investigating. Summary of geometrical theorems summarizes the proofs of concurrency of the lines that determine these centers, as well as many other proofs in geometry. See the pictures below for examples of this.

A median is each of the straight lines that joins the midpoint of a side with the opposite vertex. Where all three lines intersect is the “orthocenter”: Circumcenterconcurrency of the three perpendicular bisectors Incenterconcurrency of the three angle bisectors Orthocenterconcurrency of the three altitudes Centroidconcurrency of the three medians.

### Triangle centers: Circumcenter, Incenter, Orthocenter, Centroid

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. Check out the cases of the w and right triangles below.

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”:. It can be used to generate all of the pictures above. Where all three lines intersect is the centroidkf is also the circymcenter of mass”:.

There is an interesting relationship between the centroid, orthocenter, and circumcenter of a triangle. A height is each of the perpendicular lines drawn from one vertex to the opposite side or its extension. Triangle Cicrumcenter Where is the center of a triangle? The orthocenter is the center of the triangle created from finding the altitudes of each side. 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.

Contents of this section: The incenter is the point of intersection of the three angle bisectors. Circumcenter Draw a line called a “perpendicular bisector” at right angles to the midpoint of each side.

Centroidconcurrency of the three medians. Draw a line called a “median” from a corner to the midpoint of the opposite side. The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side.

Incenter Draw a line called the “angle bisector ” from trianlge 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”: