Proposition: 6.27: Similar Parallelogram on Half a Straight Line

(Proposition 27 from Book 6 of Euclid's “Elements”)

Of all the parallelograms applied to the same straight line, and falling short by parallelogrammic figures similar, and similarly laid out, to the (parallelogram) described on half (the straight line), the greatest is the [parallelogram] applied to half (the straight line) which (is) similar to (that parallelogram) by which it falls short.


Modern Formulation

Let a parallelogram ($\boxdot{AE}$) be bisected by the segment $\overline{DC}.$ For all points $F$ lying on the segment $\overline{DB}$, among the parallelogramic figures ($\boxdot{AF}$ like drawn above), the parallelogram $\boxdot{AD}$ has the greatest area.

Proofs: 1

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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",, 2016