# Proposition: 6.14: Characterization of Congruent Parallelograms

### (Proposition 14 from Book 6 of Euclid's “Elements”)

In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional. And those equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal. * Let $AB$ and $BC$ be equal and equiangular parallelograms having the angles at $B$ equal. * And let $DB$ and $BE$ be laid down straight-on (with respect to one another). * Thus, $FB$ and $BG$ are also straight-on (with respect to one another) [Prop. 1.14]. * I say that the sides of $AB$ and $BC$ about the equal angles are reciprocally proportional, that is to say, that as $DB$ is to $BE$, so $GB$ (is) to $BF$.

### Modern Formulation

Two parallelograms are congruent if and only if their angles and the products1 of the side lengths of an angle are in both paralleograms equal.

$$\begin{array}{rclc} \boxdot{ADBF}\cong\boxdot{BCEG}&\Longleftrightarrow&(\angle{ADB}=\angle{BGC})&\wedge\\ &&(\angle{FAD}=\angle{EBG})&\wedge\\ &&(|\overline{DB}|\cdot|\overline{BF}|=|\overline{BE}|\cdot|\overline{GB}|). \end{array}$$

Proofs: 1

Proofs: 1 2 3 4 5 6

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### References

1. The product is equivalent to Euclid's "reciprocal proportion" $$\frac{|\overline{DB}|}{|\overline{BE}|}=\frac{|\overline{GB}|}{|\overline{BF}|},$$ which is not to be confused with the concept of reciprocal ratio