# Thread: Centers of a triangle

1. ## Centers of a triangle

What exactly is the difference between circumcenter, incenter, centroid, and orthocenter? I'm having difficulty constructing the four centers in a acute triangle, right triangle, obtuse triangle, and equilateral triangle. Now is it possible to put all 4 centers in each of these triangles or no? That's my issue right now. Is it one of the listed centers per triangle or all 4 per triangle? I'm confused.

2. Hello, R3ap3r!

It takes a while to get them sorted in the brain . . .

What exactly is the difference between circumcenter, incenter, centroid, and orthocenter?
Triangles have many shapes: equilateral, isosceles, acute, right, obtuse, etc.
. . But all of them have these four centers.

Circumcenter: the common intersection of the three perpendicular bisectors of the sides.
It is equidistant from the three vertices and hence is the center
. . of the circumscribing circle.

Incenter: the common intersection the three angle bisectors of the triangle.
It is equidistance from the three sides of the triangle
. . and is the center of iscribed circle.

Centroid: the common intersection of the three medians of the triangle.
(A median joins a vertex to the midpoint of the opposite side.)
This point is the center of gravity (mass) of the triangle.

Orthocenter: the common intersection of the three altitudes of the triangle.
(An altitude is drawn from a vertex perpendicular to the opposite side.)

3. Whats the difference between altitude and perpendicular bisector?

4. Hello,

Originally Posted by R3ap3r
Whats the difference between altitude and perpendicular bisector?
The altitude is the line starting from a summit and that goes perpendicular to the opposit side.

The perpendicular bisector is the perpendicular line to a segment [AB], passing through the midpoint of this segment. Basically, every point on the perpendicular bisector will be at equal distance from A and B.

They both are perpendicular to a side of a triangle, but one will start from the summit (the altitude), the other one will just be perpendicular and going through the midpoint.

They are the same when the triangle is isoscele at the summit where the altitude comes from, or if the triangle is equilateral.

5. Originally Posted by R3ap3r
What exactly is the difference between circumcenter, incenter, centroid, and orthocenter? I'm having difficulty constructing the four centers in a acute triangle, right triangle, obtuse triangle, and equilateral triangle. Now is it possible to put all 4 centers in each of these triangles or no? That's my issue right now. Is it one of the listed centers per triangle or all 4 per triangle? I'm confused.
I've drawn an obtuse triangle PQR:

M is the circumcenter
C is the centroid
I is the incenter
O is the orthocenter

The points MCO are located allways on a straight line. (I know it under the name Euler line)

I've added the circumcircle and the incircle.

If the triangle PQR is acute all 4 points are placed in the triangle.
If the triangle is a right triangle with the right angle at Q then Q = O and M is the midpoint of PR
If the triangle is obtuse see below.