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

Then the orthocenter is also outside the triangle.

This file also has all the centers together in one picture, as well as the equilateral triangle. 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.

The line segment created by connecting these points is called the median.

Note that sometimes the circukcenter of the triangle have to be extended outside the triangle to draw the altitudes. The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side. For the centroid in particular, it divides each of the medians in a 2: The circumcenter is the center of a triangle’s circumcircle circumscribed circle.

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

You see that even though the circumcenter is outside rriangle triangle in the case of the obtuse triangle, it is still equidistant from all three vertices of the triangle. 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. Where all three lines intersect is the “orthocenter”: The angle bisectors of a triangle are each one of the lines that divide an angle into two equal angles.

The incenter is the center of the circle inscribed in the triangle. The orthocenter and circumcenter are isogonal conjugates of one another. Draw a line called the “altitude” at right angles to a side and going through the opposite corner.

Orthocenter Draw a line called the “altitude” incentter right angles to a side and going through the opposite corner. The incenter is the point of intersection of the three angle bisectors. In fact, it can be outside the triangle, as in the case of an obtuse triangle, or it can fall at the midpoint of the hypotenuse of a right triangle.

The centroid is the center of a triangle that can be thought of as the center of mass. If you have Geometer’s Sketchpad and would like to see the GSP construction of the centroid, click here to download it. For each of those, the “center” is where special lines cross, so it all depends on those lines! If you have Geometer’s Sketchpad and would like to see the GSP construction of the incenter, click here to download it.

It is the point forming the origin of a circle inscribed inside the triangle. It is the balancing point to use if you want to balance a triangle on the tip of a pencil, for example. Orthocenter, Centroid, Circumcenter and Incenter of a Triangle.

Where all three lines intersect is the centroidwhich is also the “center of mass”:. The circumcenter is the center of the circle such that all three vertices of the circle are the same distance away from the circumcenter.

In this assignment, we will be investigating 4 different triangle centers: 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. An altitude of the triangle is sometimes called the height.

### Triangle Centers

Hide Ads About Ads. The three angle bisectors of the angles of the triangle also intersect at one point – the incenter, and this point is the center of the inscribed circle inside the triangle. A median is circukcenter of the straight lines that joins the midpoint of a side with the opposite vertex.

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”: The three perpendicular bisectors of the sides of the triangle intersect at one point, known as the circumcenter – the center of the circle containing the vertices of the triangle.

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

## Triangle Centers

Summary of geometrical theorems summarizes the proofs of concurrency of the lines that determine these centers, as well as many other proofs in geometry. Circumcenter 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 construction of the circumcenter, click here to download it.

Where all three lines intersect is the circumcenter. Here are the 4 most popular ones: See the pictures below for examples of this. This page summarizes some of them. The radius of the circle is obtained by dropping a perpendicular from the incenter to any of the triangle legs. There are actually thousands of centers!

No matter what shape your triangle is, the centroid will always be inside the triangle. Contents of this section: Incenterconcurrency of the three angle bisectors. It is pictured below as the red dashed line. The circumcenter is the point of intersection of the three perpendicular bisectors. 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.

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

Remember, the altitudes of a triangle do not go through the midpoints of the legs unless you cigcumcenter a special triangle, like an equilateral triangle. The centroid is the point of intersection of the three medians.