Triangle center
The notable points of a triangle (incenter, circumcenter, centroid, and orthocenter) are fundamental concepts in geometry, each with unique properties and interesting connections, such as their alignment on the Euler line for certain types of triangles.
Instructions:
Select the points and lines you want to view by clicking on their respective boxes.
Summary
- Incenter
- Definition: It is the point of intersection of the three internal angle bisectors of the triangle.
- Properties:
- It is the center of the inscribed circle (incircle), which is tangent to all three sides of the triangle.
- It is always located inside the triangle.
- The distance from the incenter to the sides of the triangle equals the radius of the inscribed circle.
- Circumcenter
- Definition: It is the point of intersection of the three perpendicular bisectors of the sides of the triangle.
- Properties:
- It is the center of the circumscribed circle (circumcircle), which passes through all three vertices of the triangle.
- It can be inside the triangle (if it is acute), on the triangle (if it is right), or outside the triangle (if it is obtuse).
- Centroid
- Definition: It is the point of intersection of the three medians (the lines connecting each vertex to the midpoint of the opposite side).
- Properties:
- It divides each median into two segments with a \(2:1\) ratio from the vertex.
- It is the center of gravity of the triangle.
- It is always inside the triangle.
- Orthocenter
- Definition: It is the point of intersection of the three altitudes (the perpendiculars from each vertex to the opposite side or its extension).
- Properties:
- It can be inside the triangle (if it is acute), on the triangle (if it is right), or outside the triangle (if it is obtuse).
- Euler Line
- Definition: It is a straight line that passes through the circumcenter, the centroid, and the orthocenter of a triangle.
- Properties:
- In a non-equilateral triangle, these three points are collinear, and the centroid divides the segment between the circumcenter and the orthocenter in a \(2:1\) ratio.
- In an equilateral triangle, all four points coincide at one location, so there is no distinct Euler line.