Why Does Centroid of a Triangle Divides Median into 2:1?

[Math Solved] Two methods - the area method and similar triangles - are used to prove a very important property of centroid of a triangle: the centroid divides the median into two segments whose ratio is 2:1. How do we draw auxiliary lines for proving it?

#mathsolved #geometry #trigonometry

Why Does Centroid of a Triangle Divides Median into 2:1?

[leszekzdrzalik]

Tkank you. I'm From Poland, but I understand it.

[hellolello]

How do you tell the butterfly theorem? I searched it up and no results came up.

[hongningsuen]

The 3 medians divide the triangle into 6 small triangles. Let small triangles on side A have equal areas x (equal bases with equal height). Similarly let small triangles on side B and C have respective equal areas y and z. Median to side A divides triangle into 2 equal halves each consisting of 3 small triangles i.e. x + 2y = x + 2z. Hence y = z. Similarly median to side B divides triangle into 2 equal halves each consisting of 3 small triangles i.e. y + 2x = y + 2z. Hence x = z. Hence x = y = z. From this result, the 2:1 ratio of segments of median can be easily deduced.