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

	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?

Related triangle topics

General

• Triangle definition
• Hypotenuse
• Triangle interior angles
• Triangle exterior angles
• Pythagorean Theorem
• A graphical proof of the Pythagorean Theorem
• Pythagorean triples
• Triangle circumcircle
• Triangle incircle
• Triangle medians
• Midsegment of a triangle
• Triangle inequality
• Side / angle relationship

Perimeter / Area

• Perimeter of a triangle
• Area of a triangle
• Heron's formula
• Area of an equilateral triangle
• Area by the "side angle side" method
• Area of a triangle with fixed perimeter

Triangle types

• Right triangle
• Isosceles triangle
• Scalene triangle
• Equilateral triangle
• Equiangular triangle
• Obtuse triangle
• Acute triangle
• 3-4-5 triangle
• 30-60-90 triangle
• 45-45-90 triangle

Triangle centers

• Incenter of a triangle
• Circumcenter of a triangle
• Centroid of a triangle
• Orthocenter of a triangle
• Euler line

Congruence and Similarity

• Congruent triangles

Solving triangles

• Solving the Triangle
• Law of sines
• Law of cosines

Triangle quizzes and exercises

• Triangle type quiz
• Ball Box problem
• How Many Triangles?
• Satellite Orbits