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 a “perpendicular bisector” at right angles to the midpoint of each side. Let’s look at each one: Circumcenter Draw a line called a “perpendicular bisector” at right angles to the midpoint of each side. A height is each of the perpendicular lines drawn from one vertex to the opposite side or its extension. In the obtuse triangle, the orthocenter falls outside the 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. A median is each of the straight lines that joins the midpoint of a side with the opposite vertex. You can look at the above example of an acute triangle, or orthocenger below examples of an obtuse triangle and a right triangle to see that this is the case.
Contents of this section: This page summarizes some of them. If you have Geometer’s Sketchpad and would like to trianfle the GSP constructions of all four centers, click here to download it. Centroid Draw a line called a “median” from a corner to the midpoint of the opposite side.
Orthocenter, Centroid, Circumcenter and Incenter of a Triangle
The orthocenter is the point of intersection of the three heights of a triangle. If you have Geometer’s Sketchpad and would like to see the GSP construction of the incenter, click here to download it.
Summary of geometrical theorems summarizes the proofs of concurrency of the lines that determine these centers, as well as many other proofs in geometry. This file also has all the centers together in one picture, as well as the equilateral orthocenfer. 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.
Various different kinds of “centers” of a triangle can be found. It can be used to generate all of the pictures above.
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. For the centroid in particular, it divides each of the medians in a 2: Like the circumcenter, the orthocenter does not have to be inside the triangle.
Anv see the three medians as the dashed lines in the figure below. Where all three lines intersect is the centroidwhich is also the “center of mass”:.
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. 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. 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 radius of the circle is obtained by dropping a perpendicular from the incenter to any of the triangle incwnter. For each of those, the “center” is where special lines cross, so it all depends on those lines! In a right triangle, the orthocenter falls on a vertex of the triangle.
Like the centroid, the incenter is always inside the triangle.
Math Forum – Ask Dr. Math
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. If you have Geometer’s Sketchpad and would like to see the GSP construction of the centroid, click here to download it.
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”:.