1. Jun 13, 2012

### rdr3

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

If ABCD and AFED are Lambert quadrilaterals (see Fig. 3.6.16), prove, in neutral geometry, that if C, D, and E are collinear, then the geometry is Euclidean.

2. Relevant equations

I have provided a picture of the shape in question.

3. The attempt at a solution

Here are the theorems I think that are important to this proof:

By Thm 3.6.7 (just the number given in the book), the 4th angle of a Lambert quadrilateral (∠EDA and ∠CDA) is either right or acute angle.
By Thm 3.6.8 we know that the measure of the side included between 2 right angles is less than or equal to the measure of the opposite side.
By Thm 4.4.5 If, under a correspondence, the three interior angles of one triangle are congruent to the corresponding interior angles of a second triangle, then the triangles are similar.

What I did was make ABCD into 2 triangles and AFED into 2 triangles. For a ABCD, a diagonal line from E to A. And for AFED, a diagonal line from C to A. Doing this we can see by AAA Similarity Condition for Triangles, the triangles are congruent to each other. Thus, making C, D, and E collinear.

Am I on the right track? Or have the right idea?

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