Please help me with this with this geometry problem?

Question

I found this image on a website and I didn't find the resolution:

http://img208.imageshack.us/img208/8556/sssni9.png

DI=IC
Why does DG=(1/3)*DB?

Answer 1

I would solve it by mixing in a little algebra with the geometry. Think of it this way. You have an xy plane.

AD is one vertical unit
DC is one horizontal unit
DA is 1/2 horizontal unit
D is the origin.

It doesn't really matter if the two segments are the same length or not.

Now we will write the equations of the two lines, AI and DB

for DB, slope m = 1/1 = 1
and the y intercept D = 0
So the equation for DB is y = x

for AI, slope m = -1/(1/2) = -2
and the y intercept A = 1
So the equation for DB is y = -2x +1

We have two equations.

y = x
y = -2x +1

3y = 1
y = 1/3

x = y = 1/3

So
D = (0,0)
G = (1/3,1/3)
B = (1,1)

Therefore DG = (1/3)DB

Answer 2

Name the middle point of the AB side E, then AE=EB.
Connect E and C, then you can have a line parallel to AI
(because AE and IC are equal and parallel, criterion for parallelograms)
Consider a parallel to AI line passing through D and
another parallel to AI passing through B.
You have the d, b, ai and ec parallel lines.
According to the theorem of Thales,
parallel lines define proportional segments on lines cutting them.
In this case, I mean that
- since DI=IC, then DG=GF, F being the point of intersection of EC, DB (D, AI, EC parallels, cut by the dc, db lines, then
di/ic=dg/gf)
- since AE=EB, then GF=FB (AI, EC, B parallels, cut by the db, ab lines).
Consequently DG=GF=FB=DB/3

Answer 3

Extend lines AI and BC to a point of intersection which we will call P. Since BP is twice the length of AD (because I is the midpoint of DC), we know that triangle BPG is twice the size of triangle ADG. Hence GB is twice the length of DG, and you have your answer.