Centroid

圖片參考：http://www.geocities.com/abcsghk/7006121102549.jpg


As in the figure, D, E and F are mid-points of BC, AC and AB respecitvely.
The centroid is, by definition, the intersection of three medians.

By Mid-Point Theorem, EF // CB and EF = 1/2 CB.

∠EFM = ∠BCM (Corr. ∠s, EF // CB)
∠FEM = ∠MBC (Corr. ∠s, EF // CB)
∠EMF = ∠BMC (Vert. Opp. ∠s)
△EFM ～ △BCM (A. A. A.)

EF / BC = EM / MB (Corr. sides of ～△s)
1/2 = EM / MB (Since EF = 1/2 CB)
EM:MB = 1:2

You can use similar ways to show the ratio for the other medians.


Hope it helps! ^^