Centers of a Triangle

Triangle is an important polygon when we consider the geometry. It has several properties which are very specific to the triangle. In this article I have mentioned about the centers of a triangle.

1. Centroid
   
   
   
   Centroid is the point that the three medians of triangle meet together (concurrent). Following figure shows the geometry of the centroid. (median is the line, which joins the vertex and opposite mid point)
   
   Centroid is the center of gravity of a triangle and we can discuss it in the lesson "Center of Gravity".
   Here $AG:GE=BG:GF=CG:GD=2:1$
   $\therefore GE=\cfrac{1}{3}AE$
2. Circumcenter 
   
   
   Cercumcenter is the center of the circumscribed circle of a triangle. Concurrent perpendicular bisectors of each side of a triangle gives the circumcenter. 
   
   
   Here $AM=BM=CM=$ Radius of circumcircle. The angle subtend the center is equal to the double of the angle which subtend the circumference. So $AMB\measuredangle=2ACB\measuredangle=2C$.
   
   
3. Incenter
   
   
   Incenter is the center of the inscribed circle of a triangle. This is the concurrent point of three angle bisectors. The circle drawn with incenter as the center and the perpendicular length from incenter to the sides(which is equal) as the radius, is called the incircle.
   
   $ID=IE=IF=$ Radius of incircle.
4. Orthocenter
   
   
   Orthocenter of a triangle is the concurrent point of three altitudes. (altitude is the line drawn from a vertex to the opposite side, perpendicularly). Refer the following figure.
   
5. Nine point circle
   Nine point circle is the circle which goes through three mid points of sides, three feet of altitudes and three mid points of line segments from orthocenter to the vertices.
   
   Radius of circumcircle is two times the radius of nine point circle.

Note: There are other centers and lines (such as Euler line) specific to the triangle. Here I have described most important properties.