000RM.dvi

232 The classical triangle centers

The following triangle centers have been known since ancient times.
We shall adopt the following notations. LetABCbe a given triangle.
The lengths of the sidesBC,CA,ABopposite toA,B,Care denoted
bya,b,c.

8.1 The centroid

The centroidGis the intersection of the three medians. It divides each
median in the ratio2:1.

C

A

B D

F E

G

The triangleDEF is called themedialtriangle ofABC. Itisthe
image ofABCunder the homothetyh(G,−^12 ).
The lengths of the medians are given by Apollonius’ theorem:

m^2 a=

1

4

(2b^2 +2c^2 −a^2 ),

etc.

Exercise

Calculate the lengths of the medians of a triangle whose sidelengths are
136, 170, and 174.