# Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally

### (Proposition 2 from Book 6 of Euclid's “Elements”)

If some straight line is drawn parallel to one of the sides of a triangle then it will cut the (other) sides of the triangle proportionally. And if (two of) the sides of a triangle are cut proportionally then the straight line joining the cutting (points) will be parallel to the remaining side of the triangle. * For let $DE$ have been drawn parallel to one of the sides $BC$ of triangle $ABC$. * I say that as $BD$ is to $DA$, so $CE$ (is) to $EA$.

### Modern Formulation ("Side-Angle-Side" Theorem for the Similarity of Triangle)

Two triangles ($\bigtriangleup{ABC}$,$\bigtriangleup{ADE}$) are similar if and only if: * they have one congruent angle ($\angle{CAB}\cong\angle{EAD}$) * and two corresponding sides are proportional: $$\frac{|\overline{AB}|}{|\overline{AD}|}=\frac{|\overline{AC}|}{|\overline{AE}|}.$$ The same holds also if this proportion is replaced by one of the following proportionalities: $$\frac{|\overline{AD}|}{|\overline{AE}|}=\frac{|\overline{AB}|}{|\overline{AC}|},\quad\frac{|\overline{AD}|}{|\overline{DB}|}=\frac{|\overline{AE}|}{|\overline{EC}|},\quad\frac{|\overline{AB}|}{|\overline{BC}|}=\frac{|\overline{AD}|}{|\overline{DE}|}.$$

Proofs: 1

Chapters: 1
Proofs: 2 3 4 5 6 7 8 9 10 11 12

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:
@Fitzpatrick

### References

#### Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

#### Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016