ALL TRIANGLE CENTERS

ALL TRIANGLE CENTRES

There are 4 centers in a triangle. Main features of all the centers are given below.

1. orthocenter:

1. It  is the point of intersection of the altitudes to the sides from vertices.
2. Altitudes does not bisect the opposite sides.
3. Lengths of altitudes of similar triangles follow the same proportion as the corresponding sides of the triangle.

2. Incenter:

1. It is same as the center of a circle inscribed in the triangle.
2.  It is equidistant from the 3 sides of the triangle.
3. The angle bisector divides the line segments in the same proportion equal to the ratio of other two sides.
4. Lengths of the perpendiculars are in the same ratio as the sides of a similar triangle.

                                            /aoc = 90 + (/abc)/2

3. Circumcenter:

1. The perpendicular bisector of a triangle meets at a point called circumcenter of the triangle.
2. It is the center of the circle circumcribed about the triangle.
3. It is equidistant from all the vertices of triangle.
4. It can lie outside of the triangle.

4. Centroid:

1. The intersection point from the vertices to the midpoint of the sides is called centroid.
2. It divides the medians in 2:1.
3. It always lies inside the triangle.
4. It follows the similarity ratio.

coordinates of centroid are given by: