000RM.dvi

702 Routh and Ceva theorems

25.1 Routh theorem: an example ...........

Given a triangleABC,X,Y,Zare points on the side lines specified by
the ratios of divisions

BX:XC=2:1,CY:YA=5:3,AZ:ZB=3:2.

The linesAX,BY,CZbound a trianglePQR. Suppose triangleABC
has area. Find the area of trianglePQR.

2 1

5

3

3

2

A

B X C

Z

Y

P Q

R

We make use ofhomogeneous barycentric coordinateswith respect
toABC.

X=(0:1:2),Y=(5:0:3),Z=(2:3:0).

Those ofP,Q,Rcan be worked out easily:

P=BY∩CZ Q=CZ∩AX R=AX∩BY

Y =(5:0:3) Z=(2:3:0) X=(0:1:2)

Z=(2:3:0) X=(0:1:2) Y =(5:0:3)

P= (10 : 15 : 6) Q=(2:3:6) R=(10:3:6)

This means that theabsolute barycentric coordinatesofX,Y,Zare

P= 311 (10A+15B+6C),Q= 111 (2A+3B+6C),R= 191 (10A+3B+6C).

The area of trianglePQR

=

1

31 · 11 · 19

∣ ∣ ∣ ∣ ∣ ∣

10 15 6

236

1036

∣ ∣ ∣ ∣ ∣ ∣

·

=

576

6479

.