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Proposition 46 of 48 • Construction

To describe a square on a given straight line.

Given any straight line, construct a square that has that line as one of its sides. This means building a four-sided figure with all sides equal and all angles right. This construction is essential preparation for the Pythagorean theorem (Proposition 47), which requires drawing squares on the sides of a triangle.

Before You Read

Given just a line segment, can you construct a perfect square on it—all four sides equal, all four angles exactly 90°—using only compass and straightedge? It sounds straightforward, but Euclid insists on proving it from scratch because Proposition 47, the Pythagorean theorem, will need squares on all three sides of a triangle.

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A B E D C

What Euclid Is Doing

Setup: We are given a straight line AB. We must construct a square ADEB with AB as one side — that is, a quadrilateral with all four sides equal to AB and all four angles right angles.

Approach: Euclid raises a perpendicular at A, cuts off a length equal to AB along that perpendicular, then draws parallels to complete the figure. The result is a parallelogram with one right angle and all sides equal — which makes it a square. He uses Proposition 34 to show opposite sides are equal, and Proposition 29 to show all angles are right.

Conclusion: At point A, draw AD perpendicular to AB (Proposition 11). Cut off AD equal to AB (Proposition 3). Through D, draw DE parallel to AB (Proposition 31). Through B, draw BE parallel to AD (Proposition 31). Then ADEB is a parallelogram (opposite sides parallel by construction). Since angle DAB is a right angle (by the perpendicular construction) and ADEB is a parallelogram, the opposite angle DEB is also right (Proposition 34). By Proposition 29, the angles ADE and ABE are each supplementary to the right angle at A, so each is also a right angle. All four angles are right. Since ADEB is a parallelogram, AB = DE and AD = BE (Proposition 34). But AD = AB (by construction), so all four sides are equal. ADEB is a square on AB. ✓

Key Moves

  1. Given: straight line AB.
  2. Draw AD perpendicular to AB at point A (Proposition 11).
  3. Cut off AD equal to AB (Proposition 3).
  4. Through D, draw DE parallel to AB (Proposition 31).
  5. Through B, draw BE parallel to AD (Proposition 31).
  6. ADEB is a parallelogram with right angle at A.
  7. Opposite angles are equal (Proposition 34): angle DEB is also right.
  8. Adjacent angles are supplementary (Proposition 29): angles ADE and ABE are right.
  9. Opposite sides are equal (Proposition 34): AB = DE and AD = BE. Since AD = AB, all sides are equal.
  10. ADEB is a square on AB ✓

Try It Yourself

Draw a segment and construct a perpendicular at one endpoint (as in Proposition 11). Cut off an equal length along the perpendicular, then draw two parallels to close the figure. Check all four corners with a right-angle test—do they all come out exactly 90°, and are all four sides truly equal?

Proof Challenge

Available Justifications

1.

Given: Line segment AB. Draw a line at right angles to AB from point A.

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2.

On the perpendicular from A, cut off AD equal to AB.

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3.

Through D draw DE parallel to AB, and through B draw BE parallel to AD. ADEB is a parallelogram.

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4.

In parallelogram ADEB, AB = DE and AD = BE (opposite sides of a parallelogram are equal).

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5.

Since AD = AB, all four sides AB = BE = ED = DA are equal.

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6.

Since AD is parallel to BE and AB is a transversal, angle DAB + angle ABE = two right angles. But angle DAB is a right angle, so angle ABE is also a right angle. Opposite angles are equal, so all angles are right angles. ADEB is a square.

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Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 46.

  • ✓Lesson Plan (prop-46-lesson-plan.pdf)
  • ✓Student Worksheet (prop-46-worksheet.pdf)
  • ✓Answer Key (prop-46-answer-key.pdf)
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Why It Matters

This proposition provides the construction that the Pythagorean theorem (Proposition 47) depends on. Without the ability to construct a square on a given line, you cannot even state 'the square on the hypotenuse' — let alone prove anything about it. Proposition 46 is the last tool Euclid needs before the grand climax of Book I.

Modern connection: Constructing a square on a segment is equivalent to computing the square of a length — the geometric version of the algebraic operation x squared. This construction underpins the entire theory of quadratic magnitudes in Greek mathematics and is the geometric foundation for what we now call the L2 norm (Euclidean distance).

Historical note: Euclid's definition of a square (Definition 22) requires all sides equal and all angles right. This construction proves that such a figure actually exists and can be built with straightedge and compass — a non-trivial fact that Euclid is careful to establish before using squares in the Pythagorean theorem.

Discussion Questions

  • The construction uses a perpendicular and then parallels. Could you construct a square differently — say, by constructing four equal segments and four right angles directly? Would that be simpler or more complicated?
  • Why does Euclid need to prove all four angles are right? If one angle of a parallelogram is right, must the others be right too? Where is this fact established?
  • This proposition constructs the square but says nothing about its area. How does Euclid handle 'the area of a square' without a numeric area formula?
Euclid's Original Proof
On a given straight line to describe a square.

Let AB be the given straight line; thus it is required to describe a square on the straight line AB.

Let AC be drawn at right angles to the straight line AB from the point A on it, [I.11]

and let AD be made equal to AB; [I.3]

through the point D let DE be drawn parallel to AB, [I.31]

and through the point B let BE be drawn parallel to AD. [I.31]

Therefore ADEB is a parallelogram;

therefore AB is equal to DE, and AD to BE. [I.34]

But AB is equal to AD;

therefore the four straight lines BA, AD, DE, EB are equal to one another;

therefore the parallelogram ADEB is equilateral.

I say next that it is also right-angled.

For, since the straight line AD falls upon the parallels AB, DE, the angles BAD, ADE are equal to two right angles. [I.29]

But the angle BAD is right;

therefore the angle ADE is also right.

And in parallelogrammic areas the opposite sides and angles are equal to one another; [I.34]

therefore each of the opposite angles ABE, BED is also right.

Therefore ADEB is right-angled.

And it was also proved equilateral.

Therefore it is a square; and it is described on the straight line AB.

(Being) what it was required to do.

What's Next

With squares constructible on any segment, Euclid has every tool he needs. Proposition 47—the Pythagorean theorem—builds squares on all three sides of a right triangle and proves that the two smaller squares together equal the large one. Everything in Book I has been climbing toward this single result.