# Affine Geometric Problem

1. May 11, 2009

### Horse

1. The problem statement, all variables and given/known data

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 [Broken].

2. Relevant equations

AC = 2 * CB

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

where [ACD] and [CDB] are areas.

3. 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.

Last edited by a moderator: May 4, 2017
2. May 11, 2009

### Horse

Picture

#### Attached Files:

• ###### geo.JPG
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Last edited by a moderator: May 4, 2017
3. May 11, 2009

### Staff: Mentor

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

$\angle$ACD = $\angle$CDM - alt. interior angles cut by transversal
$\angle$MDB = $\angle$DAC - complements of congruent angles are congruent
$\angle$ABC = $\angle$ACD - 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.

4. May 12, 2009

### Horse

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

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