Graphics Reference
In-Depth Information
Y
10
200
r
(
x
P
, y
P
)
X
10
Fig. 11.18.
The inscribed circle for a triangle.
C
1
2
B
¢
F
2
A
¢
E
1
D
A
1
2
B
C
¢
Fig. 11.19.
Triangle ∆
ABC
with sides divided in the ratio 1:2.
Therefore, the inscribed circle has a radius of 2.929 and a centre with coordi-
nates (2.929, 2.929).
Let's explore another example where we determine the barycentric coor-
dinates of a point using virtual mass points.
Figure 11.19 shows triangle ∆
ABC
where
A
,B
and
C
divide
BC
,
CA
and
AB
respectively, in the ratio 1:2. The objective is to find the barycentric
coordinates of
D
,
E
and
F
, and the area of triangle ∆
DEF
as a proportion
of triangle ∆
ABC
.
We can approach the problem using mass points. For example, if we assume
D
is the centroid, all we have to do is determine the mass points that create
this situation. Then the barycentric coordinates of
D
are given by (11.8). We
proceed as follows.
The point
D
is on the intersection of lines
CC
and
AA
. Therefore, we
begin by placing a mass of 1 at
C
. Then, for line
BC
to balance at
A
amass
of 2 must be placed at
B
. Similarly, for line
AB
to balance at
C
amassof4
must be placed at
A
. This configuration is shown in Figure 11.20.
