In the study of triangles, the barycenter, the orthocenter, the incenter and the circumcenter are points of great importance. importance, because each one of them brings properties and characteristics that help the resolution of several problems.
These points, known as notable points, are determined by crossing a set of lines, known as cevian lines. As a triangle has three sides and three vertices, every triangle has three of each of these lines.
Barycenter
The barycenter is the meeting point (intersection) between the three medians of a triangle. Remember that the median is the segment that runs from one vertex to the middle of the opposite side.
One property of the barycenter is that it divides the median into two parts, where the smaller one is equal to 1/3 of the median itself.
Another interesting property of the barycenter is that it determines the center of mass, or gravity, of the triangle.
orthocenter
The orthocenter is the meeting point (intersection) between the three heights of a triangle. Remember that height is the segment that goes from a vertex to the opposite side, making 90°.
The orthocenter can also be on the triangle, if it is a rectangle, or outside, if it is an obtuse triangle.
incenter
The incenter is the meeting point (intersection) between the three bisectors of a triangle. A bisector is a segment that divides an angle in half, that is, determines two equal angles.
The incenter is also the center of the inscribed circle (which is inside) the triangle. In the image above, it is the dotted circumference.
The distance between the incenter and the sides of the triangle is the same for all three sides. This distance is exactly the radius of this circle.
The incenter is always inside the triangle, regardless of the shape of the triangle, since it is the center of the inscribed circle.
circumcenter
It is the meeting point (intersection) between the three bisectors. A bisector is a line that cuts a segment at its midpoint, with an angle of 90°.
The circumcenter is the center of the circumscribed circle of the triangle. The three vertices of the triangle belong to this circle. For this reason, the vertices are the same distance from the circumcenter, and this distance is the radius of the circle itself.
It is important to note that the circumcenter can be outside the triangle, or even on the triangle. In the example above the triangle is acute (three angles less than 90°) and the circumcenter is in the triangle.
If the triangle is rectangle, the circumcenter will be on one side of the triangle.
If the triangle is obtuse, the circumcenter will be outside the triangle.
Notable points and cevians
As each notable point of a triangle is formed by crossing the cevians, this table helps to distinguish each one.
| notable point | ceviana |
|---|---|
| barycenter | medians |
| orthocenter | heights |
| incenter | bisectors |
| circumcenter | bisectors |
Height, Median, Bisector, and Bisector in a Triangle
These segments are important in the study of geometry and triangles. Identify these four segments in the triangle in the image below.
The is the height;
B is the bisector;
w is median;
d is the mediator.
Learn more about triangles at:
- Triangle: all about this polygon
- Classification of Triangles
- Exercises on triangles explained
- Similarity of Triangles
- Triangle Perimeter
ASTH, Rafael. Notable points of a triangle: what they are and how to find them.All Matter, [n.d.]. Available in: https://www.todamateria.com.br/pontos-notaveis-de-um-triangulo/. Access at:
See too
- Exercises on triangles explained
- bisector
- Triangle: all about this polygon
- Bisector
- Similarity of Triangles
- quadrilaterals
- Isosceles Triangle
- 8th grade math exercises
