Triangle Centers - Overview

Thousands of years ago, when the Greek philosophers were laying the first foundations of geometry, someone was experimenting with triangles. They bisected two of the angles and noticed that the angle bisectors crossed. They drew the third bisector and surprised to find that it too went through the same point. They must have thought this was just a coincidence. But when they drew any triangle they discovered that the angle bisectors always intersect at a single point!   This must be the 'center' of the triangle. Or so they thought.

After some experimenting they found other surprising things. For example the altitudes of a triangle also pass through a single point (the orthocenter). But not the same point as before. Another center! Then they found that the medians pass through yet another single point. Unlike, say a circle, the triangle obviously has more than one 'center'.

The points where these various lines cross are called the triangle's points of concurrency.

Some triangle centers

There are many types of triangle centers. Below are four common ones. There is a page for each one. Click on the link to probe deeper.
Incenter
Located at intersection of the angle bisectors.

See Triangle incenter definition and
How to Construct the Incenter of a Triangle
Circumcenter
Located at intersection of the perpendicular bisectors of the sides

See Triangle circumcenter definition and
How to Construct the Circumcenter of a Triangle
Centroid
Located at intersection of the medians

See Triangle centroid definition
and Constructing the Centroid of a Triangle.
Orthocenter
Located at intersection of the altitudes

See Triangle orthocenter definition
and Constructing the Orthocenter of a Triangle.

In the case of an equilateral triangle, the incenter, circumcenter and centroid all occur at the same point.

How many centers does a triangle have?

Lots. Over time mathematicians have found many more. Some with names like 'Apollonius Point' and 'Symmedian point'. Many are exotic and beyond the scope of this volume, but you might find this huge list interesting: http://mathworld.wolfram.com/topics/TriangleCenters.html

Related triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises

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