# Parallel Lines and Similar Triangles:

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

### Explanation

The problem is: in ΔABC draw lines BD and CE with D on AC and E on AB. Let F on AB and G on AC be such that DF||CE and EG||BD. Then FG||BC.

Let K be the intersection of DF and EG.

We may detect several pairs of similar triangles and derive several proportions. We need just two. Since triangles ABD and AEG,

AE/AB = AG/AD.

Since triangles ACE and ADF are similar, we have also

AF/AE = AD/AC.