# Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram

### (Proposition 9 from Book 6 of Euclid's “Elements”)

To apply a parallelogram, equal to a given rectilinear figure, to a given straight line, (the applied parallelogram) overshooting by a parallelogrammic figure similar to a given (parallelogram). * Let $AB$ be the given straight line, and $C$ the given rectilinear figure to which the (parallelogram) applied to $AB$ is required (to be) equal, and $D$ the (parallelogram) to which the excess is required (to be) similar. * So it is required to apply a parallelogram, equal to the given rectilinear figure $C$, to the given straight line $AB$, overshooting by a parallelogrammic figure similar to $D$. ### Modern Formulation

This proposition is a geometric solution of the quadratic equation1 $x^2 + \alpha\,x -\beta = 0.$

Proofs: 1

Proofs: 1

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### References

1. Here, $x$ is the ratio of a side of the excess to the corresponding side of figure $D$, $\alpha$ is the ratio of the length of $AB$ to the length of that side of figure $D$ which corresponds to the side of the excess running along $AB$, and $\beta$ is the ratio of the areas of figures $C$ and $D$. Only the positive root of the equation is found (translator's note).