How to Determine the Geometric Version of the Golden Mean (Ratio or Proportion)

Step-by-Step Guide

1. 1

   Step 1: Let AB be the given straight line; thus it is required to cut AB so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.

   46[; let AC be bisected at the point E, and let BE be joined; let CA be drawn through to F, and let EF be made equal to BE (as by radius); let the square FH be described on AF, and let GH be drawn through to K.,, But EF is equal to EB; therefore the rectangle CF, FA together with the square on AE is equal to the square on EB. ,,,,,,,
   - quod erat faciendum, "that which was to have been done".
   
   The Greek loses something in the translation; it means "precisely what was required to be done"., Book II, Proposition
   11.
   
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2. 2

   Step 2: Let the square ABDC be described on AB [see Proposition I.

3. 3

   Step 3: Say that AB has been cut at H so as to make the rectangle contained by AB

4. 4

   Step 4: BH equal to the square on AH.

5. 5

   Step 5: since the straight line AC has been bisected at E

6. 6

   Step 6: and FA is added to it

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   Step 7: the rectangle contained by CF

8. 8

   Step 8: FA together with the square on AE is equal to the square on EF.

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   Step 9: But the squares on BA

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    Step 10: AE are equal to the square on EB

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    Step 11: for the angle at A is right ; therefore the rectangle CE

12. 12

    Step 12: FA together with the square on AE is equal to the squares on BA

13. 13

    Step 13: Let the square on AE be subtracted from each; therefore the rectangle CF

14. 14

    Step 14: FA which remains is equal to the square on AH.

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    Step 15: Now the rectangle CF

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    Step 16: FA is FK

17. 17

    Step 17: for AF is equal to FG; and the square on AB to AD; therefore FK is equal to AD.

18. 18

    Step 18: Let AK be subtracted from each; therefore FH which remains is equal to HD.

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    Step 19: And HD is the rectangle AB

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    Step 20: BH for AB is equal to the square on HA.

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    Step 21: Therefore the given straight line AB has been cut at H so as to make the rectangle contained by AB

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    Step 22: BH equal to the square on HA.

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    Step 23: Q.E.F.

24. 24

    Step 24: Review the chart of the Golden Mean from Euclid's Elements.

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    Step 25: 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

26. 26

    Step 26: Geometric and/or Trigonometric Art

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    Step 27: Charting/Diagramming and Algebraic Formulation.

Detailed Guide

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