Proving Proportionality of Areas with Affine Geometry

[Horse]

Homework Statement

Show that the ratio of areas is proportional to the sides squared:

\frac{[ACD]}_{[CDB]} is proportional to \frac{AC^2}_{CB^2}

Please, see the picture: http://dl.getdropbox.com/u/175564/geo_henry.JPG .

Homework Equations

AC = 2 * CB

\frac{AD}_{DB} is propotional to \frac{[ACD]}_{[CDB]}

where [ACD] and [CDB] are areas.

The Attempt at a Solution

I was unable to prove the relation with pythagoras, so I feel an easier solution. Perhaps, you can prove it somehow with affine geometry.

 

[Horse]

Picture

Please, see the picture here: http://dl.getdropbox.com/u/175564/geo_henry.JPG . Or download the attachment.

 

[Mark44]

The basic idea is that if two triangles are similar (same corresponding angles), their corresponding sides will be proportional.

\angleACD = \angleCDM - alt. interior angles cut by transversal
\angleMDB = \angleDAC - complements of congruent angles are congruent
\angleABC = \angleACD - complements of congruent angles are congruent

The statements above show that triangle ACD is similar to triangle CDB.
[ACD] = 1/2 * AD * CD
[CDB] = 1/2 * DB * CD

The two equations above show the proportionality you want.

 

[Horse]

The statements above show that triangle ACD is similar to triangle CDB.
[ACD] = 1/2 * AD * CD
[CDB] = 1/2 * DB * CD

The two equations above show the proportionality you want.

I am sorry of the blurry image. I wanted to know why the relation is true:

\frac{AC^2}_{CB^2} is proportional to \frac{[ACD]}_{[CDB]}