Describe a Square on a Given Line AB

Euclid, in his treatise of geometry, "Elements", gave in Book I Proposition XLVI, the method "On a given line (AB) to describe a square." You will learn in this article to make a Euclidean square.

Steps

The Tutorial

1. Erect AD at right angles to AB [XL.], and make it equal to AB [III.].
2. Draw DC parallel to AB through D d[XXXI.], and draw BC through B parallel to AD, then AC is the square required.
3. Demonstrate that because AC is a parallelogram, AB is equal to CD [XXXIV.]; and AB is equal to CD [XXXIV.]; and AB is equal to to AD (by construction); therefore AD is equal to CD, and AD is equal to BC [XXXIV.]. Hence the four sides are equal; therefore AC is a lozenge, and the angle A is a right angle. Therefore AC is a square (Def. XXX.).

Supporting Propositions

1. The supporting Propositions are provided above, in reverse Proposition order (not by order of mention):
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

Helpful Guidance

1. Make use of helper articles when proceeding through this tutorial:
   • See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
   • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

Things You'll Need

• Sharp pencil and paper
• Rule
• Compass
• Microsoft Excel, or an application like it.
• Handy reference to Euclid's "Elements" would be nice.

Related Articles

• Create Higher Exponential Powers Geometrically
• Find the Longest Internal Diagonal of a Cube
• Center a Circle
• Multiply and Divide Geometrically Like Mother Nature
• Prove the Acute Rule, Book II Prop. 13 of Elements
• Prove the Obtuse Rule, Book II Prop. 12 of Elements
• Prove the Intersecting Chords Theorem of Euclid
• Place a Line Equal to a Given Line at an Extreme Point
• Use Random Cut Theorem and Simple Probability
• Do Garfield's Proof of the Pythagorean Theorem
• Determine the Mean Proportion or Square Root Geometrically
• Determine the Geometric Version of the Golden Mean (Ratio or Proportion)
• Determine Numeric Golden Mean from Geometric Version
• Determine a Line = to Square Root of 3 Geometrically

Sources and Citations

• The First Six Books of the Elements of Euclid by John Casey; public domain resource - this eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at Gutenberg.
• http://www.britannica.com/EBchecked/topic/194880/Euclid

See also

• Determine the Mean Proportion or Square Root Geometrically
• Convert a Quadratic Formula to Roots Form by Completing the Square
• Determine the Geometric Version of the Golden Mean (Ratio or Proportion)
• Use Random Cut Theorem and Simple Probability
• Determine Numeric Golden Mean from Geometric Version
• Multiply and Divide Geometrically Like Mother Nature
• Modify a Brief Linear Regression Model in Excel
• Find the Longest Internal Diagonal of a Cube
• Learn the Different Equations for a Line
• Form a Perfect Right Angle Using Just String

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What links here

• Do Garfield's Proof of the Pythagorean Theorem
• Prove the Intersecting Chords Theorem of Euclid
• Prove the Pythagorean Theorem
• Determine the Geometric Version of the Golden Mean (Ratio or Proportion)
• Use Random Cut Theorem and Simple Probability
• Determine Numeric Golden Mean from Geometric Version
• Center a Circle