### INCENTER CIRCUMCENTER ORTHOCENTER AND CENTROID OF A TRIANGLE PDF

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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You see the three medians as the dashed lines in the figure below. It is the point forming the origin of a circle inscribed inside the triangle.

Sorry I don’t know how to do diagrams on this site, but what I mean by that is: Thus, the radius of the circle is the distance between the circumcenter and any of the triangle’s three vertices. If you have Geometer’s Sketchpad and would like to see the GSP construction of the circumcenter, click here to download it. Like the centroid, the incenter is always inside the triangle.

The orthocenter is the center of the triangle created from finding the altitudes of each side. Incenterconcurrency of the three angle bisectors. 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. Circumcenterconcurrency of the three perpendicular bisectors Incenterconcurrency of the three angle bisectors Orthocenterconcurrency of the three altitudes Centroidconcurrency of the three medians.

### Math Forum – Ask Dr. Math

The angle bisectors of a triangle are each one of the lines that divide an angle into two equal angles. For the centroid in particular, it divides each of the medians in a 2: Draw a line called the “altitude” at right angles to a side and going through the opposite corner. The orthocenterthe centroid and the circumcenter of a non-equilateral triangle are aligned ; that is to say, they belong to the same icenter line, called line of Euler.

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

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 incenter is the last triangle center we will be investigating.

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

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

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.

Various different kinds of “centers” of a triangle can be found.

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

Note that sometimes the edges of the triangle have to be extended outside the triangle to draw the altitudes. Let’s look at each one: 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 circumcenter. The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side.

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.

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”: It is the balancing point to use if you want to balance a triangle on the tip of a pencil, for example.

Centroidconcurrency of the three medians. SKIP to Assignment 5: The incenter is the point of intersection of the three angle bisectors. A height is each of the perpendicular lines drawn from one vertex to the opposite side or its extension.