1. propERTIES OF TRIANGLES
Perpendicular bisector --- Circumcenter
The circumcenter is the point where the perpendicular bisectors of the triangle's sides converge. The three perpendicular bisectors are the lines that cross each side of the triangle at right angles exactly at their midpoint.
The circumcenter is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices. See Circumcenter of a Triangle for more. |
Angle bisector --- Incenter
The incenter is the point where the three angle bisectors of a triangle meet. The incenter is also the center of the triangle's incircle - the largest circle that will fit inside the triangle.
The triangle's incenter is always inside the triangle. |
Height --- Orthocenter
The orthocenter is the point where the three altitudes of the triangle converge. The three altitudes are the lines that pass through a vertex and are perpendicular to the opposite side. The orthocenter is not always inside the triangle. If the triangle is obtuse, it will be outside. To make this happen the altitude lines have to be extended so they cross.
See Orthocenter of a Triangle for more. |
Median --- Centroid
The centroid is the point where the three medians converge. The medians are the lines joining a vertex to the midpoint of the opposite side.
Each median divides the triangle into two smaller triangles of equal area. The centroid is exactly two-thirds the way along each median. The centroid of a triangle is the point through which all the mass of a triangular plate seems to act. Also known as its 'center of gravity' , 'center of mass' , or barycenter. See Centroid of a Triangle for more. |
Euler line
In any triangle, the centroid, circumcenter and orthocenter always lie on a straight line, called the Euler line.
In the 18th century, the Swiss mathematician Leonhard Euler noticed that three of the many centers of a triangle are always collinear, that is, they always lie on a straight line. This line has come to be named after him - the Euler line. (His name is pronounced the German way - "oiler"). The three centers that have this surprising property are the triangle's centroid ,circumcenter and orthocenter.
Another interesting fact is that in an equilateral triangle, where all three sides have the same length, all three centers are in the same place.
Fonts:
http://www.dibujotecnico.com/saladeestudios/teoria/gplana/triangulos/Elementos_notables.php
http://gaussianos.com/los-centros-del-triangulo-incentro-baricentro-circuncentro-y-ortocentro/
http://www.mathopenref.com/
http://www.dibujotecnico.com/saladeestudios/teoria/gplana/triangulos/Elementos_notables.php
http://gaussianos.com/los-centros-del-triangulo-incentro-baricentro-circuncentro-y-ortocentro/
http://www.mathopenref.com/