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### Course: Geometry (all content)>Unit 4

Lesson 6: Medians & centroids

# Centroid & median proof

Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median). Created by Sal Khan.

## Want to join the conversation?

• Is this really a textbook "proof"? It seems that my textbook doesn't want proofs to take the form of a set of equations. I see that this does "prove" the point, but my book wants a list of theorems and postulates and such. HELP!!!
• I know. In a real test, sadly, you will probably have to list all those theorems and postulates, even completely obvious ones, (such as the fact that there is a postulate for x=x)! Sal, is doing this proof to make it easier to understand this topic.
• Doesn't he show this in a previous video as well?
• Yes, Triangle Medians and Centroid. He solve it with a three dimensional plot (x,y,z) axes. The video following that showed how to solve it in 2D which is more difficult.
• at sal says " i have drawn an Arbitrary triangle" what's an Arbitrary Triangle?
• an Arbitrary triangle is a triangle that has no definite side lengths, no definite angles, and the vertices have no definite position. In other words, it is equally likely to be ANY POSSIBLE TRIANGLE.
• Why is the centroid known as the center of gravity ? Why isn't it the circumcenter, incenter, or any other point of concurrency? Why the centroid?
• Because the centroid is the physical center of gravity. If you had a paper triangle, you could balance it on a pencil if you put the pencil under the centroid.
• But why is triangle AGE is twice the size of triangle ABG? Is there a way to prove it, and not speculate?
• Is there a relationship between the Circumcenter, the Inradius and the Centroid of a triangle? Would these three points all be the same in an Equilateral Triangle?
(1 vote)
• Yes, all three points would be the same in an equilateral triangle. If the triangle was not equilateral, then the points would fall on the same line, known as the Euler line.
• How do we know for sure that the area for all 6 triangles inside the bigger triangle are equal?
• (1 vote)
• At , how do we know that the two blue triangles together have twice the area of the orange triangle? We don't know the relationship between them other than the height, but the height doesn't even matter when you split of the blue triangles.
• what would be the reasons for each step
(1 vote)
• so what's the difference between a median and a perpendicular bisector?
(1 vote)
• Median - A line segment that joins the vertice of a triangle to the midpoint of opposite side.
Angle bisector - A line segment that divides an angle of a triangle into two equal angles.
Perpendicular bisector - A line segment that makes an angle of 90 deg (right angle) with the side of a triangle.

The common point where the medians intersect is the centroid.
The common point where the angle bisectors intersect is the incenter.
The common point where the perpendicular bisectors intersects is the circumcenter.
(1 vote)