Question

define 2/1 theorem.
prove it.

Answer 1

The centroid of a triangle divides each median in the ratio 2:1

The Theorem

D, E, F are mid-points of BC, CA, AB.

AD, BE and CF are medians.

The medians cut each others are centroid G .

We need to show that:

AG : GD = BG : GE = CG : GF = 2 : 1

Simple Proof

Reflect the triangle along AC, you can get a diagram below:

ABCB1 is a parallelogram.

BEB1 is a straight line .

Since CD = AD1 and CD // AD1,

DCD1A is a parallelogram. (opposite sides equal and parallel.)

\ DG // CG1

Since BD = DC and DG // CG1 \ BG = GG1 (intercept theorem)

BG : GG1 = 1 : 1

Since GE = EG1 , BG : GE = 2 : 1.

Answer 2

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