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.

Incenter |
Located at intersection of the
angle bisectors.
See |

Circumcenter |
Located at intersection of the perpendicular bisectors of the sides See |

Centroid |
Located at intersection of the medians |

Orthocenter |
Located at intersection of the altitudes |

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

- Triangle definition
- Hypotenuse
- Triangle interior angles
- Triangle exterior angles
- Triangle exterior angle theorem
- Pythagorean Theorem
- Proof of the Pythagorean Theorem
- Pythagorean triples
- Triangle circumcircle
- Triangle incircle
- Triangle medians
- Triangle altitudes
- Midsegment of a triangle
- Triangle inequality
- Side / angle relationship

- 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

- 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

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

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