# The centers of triangle:Centroid, Orthocenter, Circumcenter and Incenter

### Centroid:

• Centroid is the point of intersection of the three medians of the triangle.
• Median is a line joining the vertex of a triangle with the midpoint of the opposite side.

The figure below explains in detail the concept of centroid and medians: Points to remember under Centroid:

• A triangle has three medians.
• If the vertices A, B and C are denoted by the ordered pairs A(x1,y1), B(x2,y2) and C(x3,y3) then the coordinates of the centroid are given by,
G=(x1+x2+x3)/3 , (y1+y2+y3)/3
• A Centroid G of a triangle divides every median in the ratio 2:1. i.e,GC:GMc::2:1 and GB:GMB::2:1 and GA:GMA::2:1.
• Centroid of a triangle always lies inside the triangle despite the shape of the triangle. That is, Centroid lies inside the triangle for all Acute, Obtuse and Right angled triangles.

### Orthocenter:

• Orthocenter is the point of intersection of the altitudes or heights of a triangle.
• Altitude or Height is the perpendicular line drawn from a vertex to the opposite side (in case of an acute triangle) or the extension of the opposite side (in case of an obtuse triangle).

The figures below clearly explain the concept of Orthocenter and Altitudes:   Points to remember under Orthocenter:

• A triangle has three altitudes.
• Orthocenter of an acute angled triangle lies inside the triangle where as orthocenter of an obtuse angled triangle lies outside the triangle. Orthocenter of a right angled triangle lies on the vertex of the triangle at the right angle.
• For a triangle with vertices A(0,0), B(x1,y1), C(x2,y2) the Orthocenter is given by
G=-(y1-y2)(x1x2+y1y2)/x1y2-x2y1 , (x1-x2)(x1x2+y1y2)/x1y2-x2y1

### Circumcenter:

• Circumcenter is the point of intersection of the perpendicular bisectors of the three sides.
• Perpendicular bisector drawn to a side refers to the line passing through the midpoint of the line and is perpendicular to the line.

The below figures clearly explain the concet of perpendicular bisectors and the circumcenter:   Points to remember under Circumcenter:

• A triangle has three perpendicular bisectors.
• A Circle drawn with the circumcenter as the center and the distance of the circumcenter with any one of the vertices as radius is called a circumcircle.
• A circumcircle touches all the three vertices of the triangle.
• Circumcenter of an acute angled triangle lies inside the triangle.
• Circumcenter of an obtuse angled triangle lies outside the triangle.
• Circumcenter of a right angled triangle lies at the midpoint of the hypotenuse.
• To find Circumcenter of a triangle, find the distance of the circumcenter (h,k) from the three vertices. This gives two linear equations in h and k. On solving these two the point h,k and hence the circumcenter (h,k) are obtained.

### Incenter

• Incenter is the point of intersection of the angle bisectors drawn to all the three vertices of the triangle from their opposite sides.
• An angle bisector is a line drawn to a vertex from the opposite side such that it divides the angle at the vertex in to two equal halves.   Points to remember under Incenter:

• A triangle has three angle bisectors.
• A Circle drawn with the incenter as the center and the perpendicular distance of the incenter with any one of the sides as radius is called an incircle.
• An incenter touches all the three sides of a triangle.
• Incenter of a triangle lies inside the triangle irrespective of the type of the circle.
• For a triangle with Cartesian vertices , , and sides a, b and c, the Cartesian coordinates of the incenter are given by 