000RM.dvi

8.4 The orthocenter and the Euler line 235

8.4 The orthocenter and the Euler line

The orthocenterHis the intersection of the three altitudes of triangle
ABC. These altitudes can be seen as the perpendicular bisectors of the
antimedialtriangleXY ZofABC, which is bounded by the three lines
each passing throughA,B,Cparallel to their respective opposite sides.

O

C

A

B

X

Z Y

H G

XY Zis the image of triangleABCunder the homothetyh(G,−2).
It follows thatHis the image ofOunder the same homothety. We
conclude thatO,G, andHare collinear, andOG:GH=1:2.
The line containingO, G,His the famous Euler line of triangle
ABC.