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
∣ ∣ ∣ ∣ ∣ ∣
·
=