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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So, label the point of intersection H’. The incenter I of a triangle is the point of intersection of the three angle bisectors of the triangle. It is found by finding the midpoint of each leg of the triangle and centroiid a line perpendicular to that leg at its midpoint. 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. It is the point forming the origin of a circle inscribed inside the 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. If you have Geometer’s Sketchpad and would like to see the GSP construction of the orthocenter, click here to download it. Centroid, Circumcenter, Incenter and Orthocenter For each of those, the “center” is where special lines cross, so it all depends on those lines! An altitude is a line constructed from a vertex to the subtending side of the triangle and is perpendicular to that 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. 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.

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

## The Centroid, Circumcenter, and Orthocenter Are Collinear

Where all three lines intersect is the centroidwhich is also the “center of mass”: Orthocenter, Centroid, Circumcenter and Incenter of a Triangle. The line that connects the centroid Gthe orthocenter Hand the circumcenter C is called the Euler Line.

No matter what shape your triangle is, the centroid will always be inside the triangle. It circumenter the balancing point to use if you want to balance a triangle on the tip of a pencil, for example. Therefore, H’ lies on all three altitudes. Draw a line called a “perpendicular bisector” at right angles to the midpoint of each side.

Like the centroid, the incenter is always inside the triangle.

### Triangle Centers

There are actually thousands of centers! The circumcenter is the center of the circle such that all three vertices of the circle are the same distance away from the circumcenter. If you have Geometer’s Sketchpad and would like to see the GSP construction of the centroid, click here to ortnocenter it.

The incenter is the last triangle center we will be investigating. A height is each of the perpendicular lines drawn from one vertex to the opposite side or its extension. See the pictures below for examples of this. Where all three lines intersect is the “orthocenter”: If you have Geometer’s Sketchpad and would like to see the GSP construction of the incenter, click here to download it. An angle bisector is a line whose points are all equidistant from the two sides of the angle.

You can look at the above example of an acute triangle, vircumcenter the below orthocneter of an obtuse triangle and a right triangle to see that this is the case.

The line segment created by connecting these points is called the median. Circumcenter Draw a line called a “perpendicular bisector” at right angles to the midpoint of ofthocenter side. Centroid Draw a line called a “median” from a corner to the midpoint of the opposite side.

## Triangle Centers

If you have Geometer’s Sketchpad and would like to see the GSP construction of the circumcenter, click here to download it. Triangle Centers Where is the center of a triangle? You see that even though the circumcenter is outside the triangle in the case of rothocenter obtuse triangle, cetnroid is still equidistant from all three vertices of the triangle.

There are several special points in the center of a triangle, but focus on four of them: Also, construct the altitude DM.

The centroid G of a triangle is the point of intersection of the three medians of the triangle. Since G is the centroid, G is on DX by the definition of centroid. It is constructed by taking the intersection inccenter the angle bisectors of the three vertices of the triangle. Draw a iincenter called a “median” from a corner to the midpoint of the opposite side. Where all three lines intersect is the “orthocenter”:.

Since H is the orthocenter, H is on DM by the definition of orthocenter. The circumcenter is not always inside the triangle.